HEAT

W.J. R.CALVERT

HORATIO WARD STEBBINS

1878-1933

ENGINEERING LIBRARY

HORATIO WARD STEBBINS received the A.B. degree at
the University of California in 1899, and the B.S. de-
gree at the Massachusetts Institute of Technology in
1902. After twelve years of professional practice he
entered the Department of Mechanical Engineering at
Leland Stanford Jr. University where he became Asso-
ciate Professor. He was a member of the American
Society of Mechanical Engineers, Sigma Xi, and Phi
Beta Kappa. He was an ardent student and a beloved
teacher. This book is given in memory of him.

HEAT

BY

W. J. R. CALVERT, M.A.

SOMETIME SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE
ASSISTANT MASTER AT HARROW SCHOOL

LONDON
EDWARD ARNOLD & CO.

(All rights reserved}

,*«*

ENGINEERING LIBRARY

First Printed 1922
Reprinted Ififj;*^}, i&i,:t&i 1930, 1932, 1935

Printed in Great Britain by
Butler & Tanner Ltd., Frome and London

PREFACE

The book is divided into two parts. The first Part is
intended to cover the ground of a general school education ;
that is to say, it carries the work rather beyond the standard
of the School Certificate or the Army Entrance Examination.
Several of the chapters have a supplement in smaller type,
which can be omitted without disturbing the continuity of
the treatment : it is intended that they should be omitted
by those who are still at the general education stage. The
second Part, also in the same smaller type, is intended for
those who are making Heat a main subject ; it brings the
work up to University Scholarship standard, and so is ample
for such purposes as the Higher Certificate or the Intermediate
B.Sc.

In determining the treatment to be adopted hi Part I,
I was guided by a consideration which is sometimes over-
looked. Of ihose who begin the study of Heat, a fair pro-
portion will not carry their work much beyond this elementary
stage : the subject will merely form part of their general
education. While, therefore, the treatment must form a
foundation upon which a systematic study can be built up,
it cannot be regarded as a foundation only ; it must, as far
as possible, give some indication of the scope and aims of
the subject, and the kind of problems with which it deals.
If the treatment does not do this, one has been responsible
for sending out into the world a number of " educated "
people with quite erroneous ideas. Even more heavy is
one's responsibility to those who begin with an open mind
and have not yet discovered in what direction their interest
and aptitude lie ; in the elementary portion of the subject
one is supplying them with a large portion of the material

iii

iv PREFACE

upon which they must base their decision when the time
comes for them to make it.

I have assumed that the majority of those who begin the
subject will have little or no interest in the determination
of coefficients of expansion or specific heats unless it is made
clear to them, at the outset, that objectives which appear
to them reasonable cannot be reached without dealing with
such measurements.^ In one of J. B. Biot's works there
occurs a sentence which hap always appealed to me as extra-
ordinarily appropriate. " Toutes ces choses ne peuvent
se determiner surement que par des mesures precises que
nous chercherons plus tard ; mais auparavant il fallait au
moins sentir le besoin de les chercher."

I have also rather emphasised the practical applications
because at this stage the concrete usually has a stronger
appeal than the abstract. At the same time I have tried
to keep attention fixed upon the underlying principles, and
have not shirked reasonable difficulties : the book is not
intended to be " Science made easy," and my experience
is that most beginners are prepared to make serious effort
(or at any rate can be induced to do so) if they are satisfied
that their efforts lead to a reasonable end. Tn any case
there is no Royal Road to success.

One or two points in the treatment of the book as a whole
may be mentioned. In all experimental work special atten-
tion is directed to the degree of accuracy to which the work
is likely to attain One of the objects of a scientific training
is the realisation that in the search for truth the ultimate
appeal is to experiment ; but this realisation can hardly be
expected to come to those who carry out operations without
any idea of the extent to which their results can be trusted.
For example, suppose a beginner carries out an " experi-
ment " to verify some physical law, and does not know the
precise limitations of his apparatus and method. If the
results come out (by accident) more or less as he hopes or
expects he is apt to say that he has verified the law ; if they
do not, he may take refuge in the statement that if the

PREFACE v

apparatus had beeti " more accurate " the results would
have verified the law. This is appeal to the text-book, not
to experiment, and is the reverse of scientific training ; it
might possibly be suitable preparation for a successful political
propagandist. It is true that any one who has learnt to
have a (justifiable) confidence in his own experimental work
and to weigh evidence, has already achieved considerable
development ; but the sooner he begins to learn the better.

In the same way I have tried, wherever possible, not to
speak of any effect without giving some idea of its magnitude.
It can only cause unnecessary confusion to a beginner if he
reads a statement that vapours near their saturation points
do not obey the gas laws and then finds that in calculations
on the chemical hygrometer (quite possibly the only calcula-
tions of this type which he meets) it is tacitly assumed that
they do. A sense of proportion is another useful result
of a genuine scientific training.

I have thought it worth while, even in so elementary a
book, to give references to original papers : no doubt com-
paratively few readers will look them up, but those who do
will gain a great deal. Even those who do not will at least
be able to use the dates to get some idea of the chronological
development of the subject. Perhaps also it is a check on
the writer ; the fact that the source of information is given
and that he has read these papers may help to carry convic-
tion that he is describing experiments which have been done
and not things which might have been but were not.

I have also thought it worth while to give details and
dimensions in the case of many experiments of the laboratory
or lecture type ; such details may possibly be of use in saving
time for those who are arranging a course and are not able
to give as much time as they would like to preliminary experi-
ment. In any case the information can do no harm and is
some guarantee that all the work has actually been carried
out. Many of them are the outcome of experiments which
I have made for my own information in an examination of
the degree of accuracy of certain familiar operations.

vi PREFACE

It is hardly necessary at the present day to make any
comment on the use of very simple calculus in Part II ; I
do not anticipate that many readers will find it a source of
trouble. It is true that something is to be said for the very
cumbrous treatment which avoids it, but life is short ; it
is conceivable that a clearer insight into the ideas of addition
and subtraction might result from the use of a swan pan
in the early stages of arithmetic, but I have not seen any
suggestion for its revival.

In conclusion I should like to express my gratitude to
Mr. Archer Vassall for his interest in this book and his very
great help, especially in giving me the benefit of his unrivalled
gift for detecting whether a particular presentation of a
subject will appeal to certain types of mind or not ; to
Mr. Parsons, of the Physical Laboratories, who has made a
good deal of the apparatus described, and last but not least
to many Harrovians past and present, who have been con
cerned with me in the study of this subject.

W. J. R. C.

CONTENTS

PART I

OHAP. PAGE

I INTRODUCTORY .1

II TEMPERATURE AND THERMOMETRY 8

III EXPANSION or SOLIDS . . . . .25

IV PRACTICAL CONSEQUENCES OF EXPANSION . . 41
V EXPANSION OF LIQUIDS . . . . .51

VI EXPANSION OF GASES 70

VII HE AT^ AND ITS MEASUREMENT ... 84

VIII HEAT ENGINES, THE RELATION OF HEAT AND

WORK . . . . . . 103

IX CHANGE OF STATE, SOLID TO LIQUID . . 129

X CHANGE OF STATE, LIQUID TO VAPOUR . .147

XI HYGROMETRY . . . . . . .171

XII TRANSMISSION OF HEAT, CONDUCTION . . 186

Kill TRANSMISSION OF HEAT BY CONVECTION, SOME

PRACTICAL EXAMPLES . . .201

XIV RADIATION .209

XV LAWS OF RADIATION . . . . 233

XVI ON THE MEASUREMENT OF TEMPERATURE IN

INDUSTRIAL PROCESSES . . . .242

vii

viii CONTENTS

PART II

CHAP. PAGE

XVII PROPERTIES OF GASES ... . 257

XVIII THE KINETIC THEORY OF GASES . . .266

XIX PROPERTIES OF A GAS, ENERGY CONSIDERATIONS,

LIQUEFACTION OF GASES . . . .278

XX THE GAS EQUATIONS . .294
XXI THERMODYNAMICS 306

XXII THE TRANSFORMATION OF ENERGY, THE IDEA

OF ENTROPY AND ITS APPLICATIONS . . 322

APPENDIX. QUESTIONS ..... 334
INDEX ...... . 357

PART I

CHAPTER I
INTRODUCTORY

If we place our hands on a wall upon which the sun has
been shining on a summer's day, we are conscious of a sen-
sation which we express by saying that the wall feels " hot " ;
if we pick up a lump of ice, or a frog, we are conscious of a
sensation which we express by saying that the ice (or the
frog) feels " cold."

Now it is not possible to put into. words what we mean
by any particular sensation (let the student who doubts this
try to think how he would explain to a man blind from birth
what he means when he says that an object is green ; or to
a hard-working charwoman the particular joy of a really
long drive at golf or the ascent of a difficult bit of rock) ;
but if we try to consider why the wall feels hot or the frog
cold we find that another idea comes in. For example, when
one gets into a hot bath, not only is one conscious that the
water is " hot/' but one has a feeling that something is passing
from the water to oneself which makes one feel warm. To
this " something " we give the name heat, and we try to
get a mental picture of what is going on by saying that the
water feels hot because when we get into it heat passes from
the water to us.

This idea of heat as something which can be communi-
cated is brought out more definitely if we consider the case
of a gas-ring. We are quite clear that the flame of the
gas-ring is not getting hotter, but we know that if we put a

1 B

2 HEAT

saricfc]pan\o£ cold water on the ring the water will get steadily
hatter until : ultimately it boils. The gas-ring then can go
:otorsu0>lymg this ''something" which we call heat as long
.a& if is'-aligirt'" , ;

We have here two distinct ideas : one that an object may
be in a certain state which we call " hot " or " cold " ; the
other that when an object becomes hotter something which
we call heat is communicated to it. Following up these
ideas we then go on to say that an object feels hot because
it can communicate heat to us when we touch it. It should
be noted that this last sentence is not a definition of the
condition of hotness ; the idea of hotness came first as a
result of a sensation, and the idea of heat was developed in
an attempt to picture what was happening. This condition
of an object is usually expressed in scientific work as its
temperature ; a hot object is said to be at a high temperature
and a cold one at a low temperature. These terms will be
dealt with more fully in the next chapter, while the nature
of heat is considered in Chapters VII and VIII. At the
moment it will be convenient to notice some of the effects
of heat, and then some of its sources.

Effects of Heat. — 1. One of the most common effects of
heat on a substance is that a change in size takes place. In
nearly every case a substance when heated expands ; * this
expansion may be illustrated by the following simple
experiments.

In Fig. 1, D represents a metal rod or tube (a piece of iron
gas-pipe answers very well) 6 or 8 feet long. This is carried
on two supports F, G. One end of the rod is prevented
from moving by means of a heavy weight, while the other
end is in contact with a small block of wood B, placed in
alignment with two similar blocks A and C. The rod is
heated by means of half a dozen burners, and when it is hot

' Water between 0° C. and 4° C. contracts when heated ; some
specimens of invar (Chapter IV) contract slightly. A piece of india-
rubber stretched by a weight appears to contract on heating, but this
is an indirect effect as a greater force is required to stretch the india-
rubber when warm than wnen cold.

INTRODUCTORY 3

it will be seen that B is pushed out about J to J inch beyond
the alignment of A and C. If the rod is allowed to cool it

Fio. 1.

returns to its original length, leaving a gap between its end
and the block B.

To illustrate the expansion of a liquid a flask is taken and
fitted with a cork through which
passes a narrow glass tube. The
flask is completely filled with
water (which may be coloured by
adding to it ink or a little aniline
blue to make it more easily seen)
so that the liquid stands a little
way up the tube. A piece of paper
is placed behind the tube and a
mark is made at the surface of
the water (Fig. 2). On heating
the flask, the water at first sinks
a little. This is because the flask
gets hot before the water, and as
the glass expands the capacity of
the flask increases and so the
water-level sinks. In a short
time, however, the water-level will
be seen to rise above the mark and
may ultimately reach the top of
the tube ; for when the water is
heated it expands more than the pIG< 2.

To illustrate the expansion of a gas a flask and tube similar
to that of the last experiment is taken, but the flask is simply

HEAT

CM

filled with air. It is inverted so that the end of the tube
dips under the surface of some water in a bowl (Fig. 3).
On warming the flask slightly (the heat of the hand is suffi-
cient), the air expands and bubbles of it may be seen escaping
from the end of the tube through the water. On allowing
the flask to cool the air left in the flask contracts again, and
the water will be seen to rise in the
tube, driven up by the pressure of the
atmosphere on the water in the bowl.
In general solids expand a little,
liquids more,1 and gases very much
more when heated through the same
range of temperature.

Other effects of heat are —

2. A change of state may occur, e.g.
ice may be turned to water or water
to steam.

3. In general, but not always, a
rise in temperature is caused.

These two effects are dealt with in
Chapters IX and X.

4. An effect of heat of which use is
made in the thermopile may be noticed
here. If two dissimilar metals are
joined together at one end and the
other ends are in metallic connection

either through another wire, or directly, then on warm-
ing the junction a small electric current will flow round
the circuit. This may be illustrated by taking two pieces of
wire, one iron and the other copper (ordinary 22-gauge con-
necting wire does very well), soldering them together at one
end and connecting the free ends to a sensitive 2 mirror

1 An exception is the case of a liquid very near the critical point,
where the expansion is in some cases greater than that of the gas (see
Chapter XVII).

2 The instrument used by the writer for this purpose and for use
with a thermopile is a Paul mirror galvanometer resistance 8-75 ohnis
and sensitivity 10 mm. at 1 metre from the mirror for 1 micro-ampere.

FIG. 3.

INTRODUCTORY 5

galvanometer. On heating the soldered junction witn the
flame of a match, a decided deflection of the galvanometer
will be obtained, indicating that a current is passing. This
effect is greater with some pairs of metals than others ;
antimony and bismuth form a good pair, so do platinum
and rhodium.

The thermopile is an instrument in which a number of
small bars of two metals (nearly always antimony and
bismuth) are joined in series. They are imbedded in an
insulating material (pitch) with one set of junctions pro-
jecting at one end and the other set at the other.

To
Galvanometer

FIG. 4.

Fig. 4 shows only three junctions at each end ; a usual
number is sixteen or twenty -five arranged in four or five rows.
A cap is placed over one end of the instrument covering one
set of junctions B ; on bringing a warm object near the other
face a deflection of the galvanometer is obtained. If A is
covered by the cap, and B exposed to the warm object, the
current flows in the opposite direction. It does not matter
which face is exposed so long as the other is covered, unless
it is desired that the deflection of the galvanometer should
be in one direction rather than the other. With a good
galvanometer the thermopile forms a very sensitive
instrument.

Sources of Heat. — The following sources of heat may Le
mentioned —

1. The sun.

6 HEAT

2. Mechanical action, such as friction, percussion, com-
pression.

3. Chemical action, e.g. combustion, explosion.

4. Electrical action, e.g. electric radiators, incandescent
lamps.

5. Internal heat of the earth.

6. Radio-activity.

7. Physical change of state (see Chapters IX and X).
1 and 3 are outside the scope of this book.

2. Mechanical Action. — A simple experiment to illustrate
the production of heat by friction is to take a drawing-pin
and stick it into the end of a cork which serves as a handle.
The head of the drawing pin is then rubbed briskly on a piece
of blotting paper laid on the table. In a few moments the
drawing pin will have become so hot that it is painful to
touch.

To illustrate the production of heat by percussion, take a
small piece of lead with a piece of thread or silk attached to
it by which it may be lifted. Place the lead on a sheet of
blotting paper laid on the top of a large weight which serves
as an anvil. Strike the lead vigorously several times with a
hammer. On holding the lead (by means of its thread) near
a thermopile a deflection of the galvanometer shows that the
lead has become hot : this will also be evident on touching
the lead. In the absence of blotting paper the lead would
communicate heat so quickly to the iron weight that very
little effect would be obtained.

The production of heat by compression will be familiar to
any one who has pumped up a bicycle tyre ; it can be shown
more strikingly by the old-fashioned toy known as the " fire
syringe." This is a stout glass tube closed at one end and
fitted with a well-fitting piston. To the underside of the
piston is attached a small piece of cotton wool soaked in
carbon di-sulphide, a very inflammable substance. The
piston is inserted and then driven smartly down. The heat
developed by the compression of the air causes the carbon
di-sulphide to burst into flame.

INTRODUCTORY 7

4. Electrical Action. — This may be illustrated by connecting
two or three inches of thin (No. 40 S.W.G.) copper wire to a
few (say three) accumulators. The copper wire will be seen
to become red hot, and by using more cells the wire can be
fused.

5. Internal Heat of the Earth. — It is found that below the
surface of the earth the temperature increases with the depth ;
it is notably warmer at the bottom of a deep mine than at the
surface. The increase of temperature with depth varies
considerably ; the rough figure of 1° F. per 60 or 70 feet gives
some idea of the amount. The interior of the earth near the
centre must be very hot. It is possible that some part of
this effect is connected with No. 6.

6. Radio-activity. — The radio-active substances, such as
radium, are constantly undergoing change into other sub-
stances ; in the course of this change they give out certain
types of electrical radiation and may give out an appreciable
amount of heat.1 Thus a very small quantity of radium
bromide is able to maintain itself appreciably hotter (two or
three degrees) than its surroundings. A given quantity of
radium would be half transformed in something like 2,000
years and this remainder would be half transformed in another
2,000 years, and so on. During all this time heat is being
developed.

1 Owing to absorption of some of the radiation in the material itself
(see Chapter VIII, "Relation of Heat and Work").

CHAPTER H
TEMPERATURE AND THERMOMETRY

It was pointed out in the last chapter that an object might
be in the condition known as " hot " or " cold," these terms
meaning that certain sensations would be experienced if the
object were touched ; and that this condition was referred
to in scientific work as the temperature of the body. When
we speak of a body at a high temperature, however, we
imply something more than saying that it is hot ; we imply
that there is some standard of comparison, so that we can
say quite definitely that one object is hotter than another,
and that we can state the degree of hotness referred to some
scale. We must now consider how this more complete idea
is obtained.

As long as we are dealing with the same substance, our
sense of touch will furnish us with a rough estimate of degrees
of hotness. Thus in the case of water, we speak of it as
cold, tepid, warm or hot, and these terms have some signifi-
cance.1 But when we deal with different substances the
sense of touch is altogether unreliable. For example, if we
take a piece of wood and a piece of iron and leave them on
the ground on a winter's day, we shall find that the iron
will feel colder than the wood. On the other hand, if we
put the iron and the wood into a hot oven for some time,
we shall find that the iron feels hotter than the wood. The
reason for this is that an object feels hot or cold according

1 The sense of touch is not always to be relied upon. Liquid air is
very cold and red-hot iron is very hot, but the effect of these two things
on the hand is somewhat similar ; bad blisters and much pain resulting.
Inexperienced people have been known to say that liquid air had
" burnt " them.

8

TEMPERATURE AND THERMOMETRY 9

to whether it heats or cools our hand when we touch it, and
we estimate the hotness or coldness by the rate at which
it does this, or as we have already expressed the matter, by
the rate at which heat passes into or out of our hand. This
rate depends on two things, first how hot the body is, and
secondly on whether it is a good conductor of heat, i.e.
whether heat can flow through it readily or not. Now it
happens that iron is a much better conductor of heat than
wood (see Chapter XII) ; so when we touch the iron from the
oven, it is able to heat the hand more rapidly than the wood
can and so feels hotter ; when the wood and iron are cold,
the iron as the better conductor is able to cool the hand
more rapidly than the wood can, and so feels colder.

There is, however, a perfectly definite test we can apply.
If we put a red-hot iron bar into a dish of cold water, we
know that the iron cools down, and the water becomes
warmer ; if, however, the iron bar, instead of being made
hot, had been left in a room beside the dish of water, then
on putting the iron into the water we could not detect any
change in either of them, whether we use our sense of touch
or one of the artificially sensitive instruments described
below. We can take this as our test and say that — If
when two bodies are placed in contact neither of them be-
comes hotter or colder than it was before these two bodies
were at the same temperature.1 If one of them becomes
colder, and the other warmer, they were at different
temperatures.

Thermometry.

Such a test is seldom convenient to carry out in practice,
and it only tells us whether two bodies are or are not at the
same temperature. In order to detect differences of tem-

1 This only applies if there is no chemical action between them ; if
strong sulphuric acid is added to water, both being at the same tem-
perature as registered by a thermometer, the mixture will be much
hotter than the constituents as heat is developed by the chemical action
of the acid and water.

10

HEAT

perature and express it on some scale an instrument called
a thermometer is required. In constructing thermometers,
use is made of one of the effects of heat, and the one most
commonly used is the expansion of a liquid such as mercury
or alcohol.

To Construct a Simple Thermometer. — The ther-
mometer when finished will consist of a thick- walled tube of
narrow bore, terminating at one end in a bulb. The bulb
and part of the stem are filled with mercury or amyl alcohol ;

if alcohol is used it must
be coloured, e.g. by a little
cochineal. The tube is sealed
at the top, and the space
above the liquid is free from
air. If the bulb be placed
in warm water, the liquid
will be heated up until the
water and the liquid are at
the same temperature. The
liquid will consequently ex-
pand and will rise in the
tube, and since this is of
narrow bore, a small expan-
sion will result in a consider-
able rise of the liquid. The
final position will depend on
the temperature of the water,
and will thus give us some
means of estimating how hot
the water is.

A piece of thick-walled narrow-bored tubing is selected
and a length of about 12 inches is cut off. One end of this
length is sealed and a bulb of suitable size is blown. The
tubing is then drawn out slightly close to the open end so
as to form a constriction. A small funnel is attached to the
open end by a piece of rubber tubing A, and the whole sus-
pended from a stand as shown in Fig. 5. Some of the liquid

FIG. 5.

TEMPERATURE AND THERMOMETRY 11

to be used, either mercury l or amyl alcohol is placed in the
funnel ; it will not run down into the bulb owing to the
narrowness of the tube. The bulb is gently warmed by
means of a burner, and the air in it expands, bubbles escaping
through the liquid in the funnel. The heating must be
gentle, otherwise the escaping air is apt to eject most of the
liquid from the funnel. The bulb is then allowed to cool,
and as the air contracts some of the liquid runs down the
tube and into the bulb ; it may be necessary to cool the
bulb by immersing it in a beaker of cold water, and this is
quite safe if the previous heating has not been carried far.
The process is then repeated, taking care to dry the outside
of the bulb before heating ; in this way the bulb is about
three-quarters filled with liquid.

It is not, however, possible to get rid of all the air in this
way and the final stage of the filling is done as follows. Any
remaining liquid in the funnel is emptied out, and the bulb
is then heated so that the liquid in it boils. The vapour
from the boiling liquid drives out the air and fills the space
above the liquid. The boiling is continued until the vapour
has been issuing from the funnel for some time, then the
funnel is filled up with more liquid and the heating stopped.
As the vapour in the bulb and tube cools, it condenses, and
the liquid rushes in and fills up the tube and bulb. Care is
necessary to see that there is sufiicient liquid in the funnel
to fill the bulb and tube, otherwise air will follow the liquid
and the whole process will have to be done again. It is not
always easy to get rid of the last traces of air, a small bubble
being very apt to form just under the shoulder where the
tube and bulb join. This bubble can sometimes be got into
the tube by taking the thermometer in the hand, holding it

1 Mercury should not be used by beginners, as its boiling point is so
high that accidents to the tube are rather likely to happen, and more-
over mercury vapour is not healthy. Amyl alcohol is selected as
having a boiling point higher than that of water, but not so high as to
involve intense heating. The vapour of amyl alcohol has a most
unpleasant effect on the throat and so should not be inhaled unneces-
sarily.

12 HEAT

by the tube just above the bulb, the upper end of the tube
lying along the forearm, and swinging it to and fro with a
stiff arm. The liquid being heavier than the bubble tends
to get as far away a-, possible from the point about which
it is swung and hence the bubble is displaced up the tube.
Once there, a little judicious warming may get it above the
constriction into the funnel where it can escape.

We have now got the bulb and tube full and the next
thing to do is to adjust the quantity of liquid so that it will
stand at the proper height in the tube. This will obviously
depend upon what sort of temperature the thermometer is
intended to register. In this case we shall adopt an arrange-
ment very often met with and make our thermometer capable
of registering temperatures up to, and a little above, the
boiling-point of water. The thermometer when finished will
be sealed off at the constriction ; we must therefore adjust
the amount of liquid so that when the thermometer is in
boiling water, the level of the liquid will be a little below
this point (about an inch and a half). Remove the funnel
and place the thermometer in boiling water. The liquid
expands, and as it oozes out of the open end of the tube it
is wiped off. Now remove the thermometer, dry the outside
and then warm it over a flame and drive out a little more of
the liquid. Now replace the thermometer in boiling water
and see whether the liquid stands at the proper height below
the constriction. If it does, we can proceed to the next
process, i.e. sealing off.

A blowpipe is prepared and the thermometer is once more
dried and heated until the liquid has expanded beyond the
constriction. It is then allowed to cool and the liquid care-
fully watched. As soon as it has passed back through the
constriction the blowpipe flame is at once directed on this
spot, the end of the tube drawn off and the sealing of the
tube completed. The thermometer should be set aside for
a few days before anything more is done to it, as the glass
after heating does not return to its normal size and condition
for some time.

TEMPERATURE AND THERMOMETRY 13

We have now got an instrument which will enable us to
say whether two objects are at the same temperature or not.
For if we put it in contact with one of them, it will be heated
or cooled until it is at the same temperature as the object ;
the liquid will expand or contract until the surface is in a
certain position. If it is put in contact with the other, and
the liquid comes to the same position, the two objects are
at the same temperature.1 In order that the indications of
the instrument may have some definite meaning it must be
graduated. For this purpose we must determine the position
of the liquid when the instrument is at two temperatures
agreed upon as standards. These two positions are called
the " fixed points " of the thermometer. The temperatures
are the temperature at which pure ice melts and the tem-
perature at which pure water boils under standard atmo-
spheric pressure, i.e. when the barometer stands at a height
of 760 mm. or 29-9 inches. (It is found that the boiling-
point of any liquid varies with the pressure ; see Chapter X.)
These two points are called the lower fixed point and the
upper fixed point respectively. The interval between the
fixed points is divided into a number of equal parts called
degrees, and the whole then forms a scale of temperature.
There are three scales of temperature in general use. That
used in almost all scientific work and for ordinary use on
the Continent (except the Scandinavian countries) is called
the centigrade scale. The interval between the fixed points
is divided into one hundred degrees ; the lower fixed point
is called 0° C. (nought degrees centigrade) and the upper
100° C. (one hundred degrees centigrade). The scale used
in England for ordinary non-scientific work is called the
Fahrenheit scale, after its originator. It is the oldest of the
three scales. On this, the interval between the fixed points
is divided into 180 degrees, but the lower fixed point is not
taken as 0 but 32, so that the upper fixed point is therefore

1 This is not a self-evident proposition. It depends on the experi-
mental fact that if two substances separately pass the test of equality
of temperature (given on page 9) with a third, they will pass tins test
of equality with one another when put in contact.

14

HEAT

212° F. The zero of this scale is therefore 32 degrees Fahrenheit
below the lower fixed point or freezing point. The third
scale, known as the Reaumur scale, is in less common use,
but is met with in the Scandinavian countries. The interval
between the fixed points is divided into 80 degrees ; the lower
fixed point is called 0° R. and the upper 80° R.

Thus in all three scales the fixed points represent the same
temperatures, but the interval is divided up differently, and
different numbers are assigned to the fixed points.

Determination of the Lower Fixed Point. — A large
funnel is taken and filled with small pieces of clean ice.1
Underneath the funnel is placed a dish to
catch the water which drains away as the
ice melts. The bulb of the thermometer
is immersed in the ice, which is packed
round it, as in Fig. 6, and heaped up to
such a height that when the liquid has
ceased contracting it is just visible above
the ice. A little distilled water is poured
over the ice to wash it, and the apparatus
left for a considerable time, say half an
hour. A fine scratch is made in the glass
(by means of a file) at the level of the
liquid. The apparatus is then left for a
further period and then inspected again to
see that the position of the surface of the
liquid is still at the mark. If it is, this
may be taken as the lower fixed point. If
it is not, a new mark must be made, and
tested by leaving the apparatus for a
further period to see if the liquid re-
mains there. The mark must not be
taken as the lower fixed point until it
FIG. 6. satisfies this test.

1 The ice must be clean, as the melting point is affected by soluble
impurities. It is also desirable that the water formed should be able
to drain away, although this point is sometimes disputed, but see page
133, Chapter IX.

TEMPERATURE AND THERMOMETRY

15

Determination of the Upper Fixed Point. — To determine
the upper fixed point the thermometer must be suspended in
the steam l of boiling water, and the apparatus employed
for this purpose is known as a hypsometer (Fig. 7). It
consists of a cylindrical metal vessel A to serve as a boiler,
surmounted by a tube C open at the top. C is surrounded
by a wider tube, with an outlet B on one side and a glass
U tube D containing water to serve as a manometer or pres-
sure gauge. The wider tube is pro-
vided with a lid, which has an opening
at the centre closed by a cork through
which passes the thermometer. A con-
tains water which is boiled, and the
steam passes up C, then down the
wider tube and escapes from the out- rrC

let. The thermometer must hang in
the steam and the bulb must not touch
the water nor be near enough to it to be
splashed. It must be so far down that
when the surface of the liquid comes
to its final position, it is just visible
above the cork. The water in the two
limbs of D must stand at the same
level ; when this is the case the pres-
sure of the steam is the same as that of
the atmosphere. If the water is boiled
too rapidly the steam cannot escape
fast enough from B and the gauge D
shows an increase ol pressure. This
must be remedied by heating less FIG. 7.

strongly.

When the steam has been issuing freely for some time,
say twenty minutes, a mark is made at the surface of the
liquid in the thermometer. The reading of the barometer

1 The temperature of the water may be above the true boiling point
for pure water if there are any impurities, and even if there are not the
temperature of the water may be too high (see Chapter X, page 160).

16

HEAT

is taken, and if this is not 760 mm. a correction must be
applied by means of the tables showing vapour pressures.
The mark is tested as in the case of the lower fixed point
by leaving the apparatus for some time and seeing that the
liquid is still in the same position. Having determined the
two fixed points, we can now graduate the ther-
mometer.

If we decide to graduate our thermometer on
centigrade scale, we can take a strip of paper
and mark off on it a distance equal to that be-
tween the two marks we have made on the
tube. We can then divide up this distance into
100 equal parts, and number them from 0° to
100° C.

To mount the thermometer, a strip of wood is
taken about 2 inches wide and a little longer
than the thermometer. A hole is made to ad-
mit the bulb. The thermometer is laid on the
wood and the paper scale adjusted behind it so
that the 0 comes opposite the lower fixed point
and the 100 opposite the upper. The whole is
then fixed in position by drilling two parrs of
small holes through the scale and wood on either
side of the tube and passing pieces of wire
round the tube, through the holes, and twisting
them up tightly at the back. The final product
appears as shown in Fig. 8.

This thermometer is easy to make, and is
quite suitable for some purposes. It has, how-
ever, certain serious disadvantages. The paper
scale prevents it from being put into liquids ; it
is large, which makes it cumbersome and, what
is worse, makes it take an appreciable quantity of heat from
any hot object with which it is put in contact, so that the
final temperature recorded by the thermometer is not the
original temperature of the object. Moreover it requires a
considerable time for the liquid to be heated up owing to

Fio. 8.

TEMPERATURE AND THERMOMETRY 17

the bulb being a sphere the surface of which is smaller in
comparison to the volume than that of any other shaped
object, and alcohol is a bad conductor of heat. For these
reasons a modified form is generally used the thermometer
is smaller, the bulb is cylindrical instead of spheri-
cal, and the liquid is almost always mercury.

One type of thermometer is shown in Fig. 9.
It consists of a thin tube A, terminating in a cylin-
drical bulb. A wider tube B sealed on to the
bulb as shown, serves to protect the thermometer
tube and to contain the scale which is engraved
on a flat piece of milk glass C. In some forms the
upper end of inner tube is sealed in position by
three small pieces of glass D ; in others, the
attachment of the scale to the tube (by means of
two small pieces of wire in much the same manner
as in our simple thermometer) gives it sufficient
lateral support, since the scale fits tightly into the
outer tube. Thermometers of this type are usually
fairly cheap and the scale is easily read and is
protected from any liquids in which the thermo-
meter may be immersed. Very cheap types are
supplied with paper scales, but these are not so
satisfactory. This type, known as the " sleeve "
pattern, is of German origin.

Another type of thermometer is shown in Fig. 10.
Thermometers for accurate work are always of this
type and not of the sleeve pattern. It consists of
a thick-walled tube of very narrow bore, terminating in
a cylindrical bulb. The scale is etched on the surface of
the tube itself, and in order to make this more visible,
the back of the tube is made opalescent, or in some
cases a strip of opalescent milk glass is worked into the
tube when it is made, so that the section of the tube is
as shown in Fig. 10. The bulb is blpwn separately and
sealed on to the tube. Thermometers of this type are
undoubtedly more suitable for accurate work, and are also

o

18

HEAT

the only type which can be used for high temperature
work (see below). They are also less liable to break. They
are, however, more expensive than the sleeve pattern although
cheap forms are to be obtained. These latter are usually
not very satisfactory for general work ; the scale is generally
not so easy to read as in the sleeve pattern, and this trouble
is often accentuated by the use of unsuitable
material for marking the scale, which is attacked
and dissolved by liquids such as alcohol, and so
after a time only the actual etching is left and
the scale becomes very difficult to read.

Mercury thermometers can be obtained read-
ing up to 400° C. in spite of the fact that at
ordinary pressures mercury boils at 357° C. This
is done by making use of the fact that the boil-
ing-point of a liquid is raised by pressure. The
space above the liquid, instead of being vacu-
ous, is filled with nitrogen under
^^ a pressure of three or four atmo-
spheres . In order to leave sufficient
room for this gas when the mer-
cury rises nearly to the top of the
tube, the upper part of the bore
is enlarged to a small bulb, as
shown in Fig. 11.

This arrangement is usually
adopted even for low temperature
thermometers as a safeguard
against bursting the thermometer
by accidental overheating ; it

forms a small reservoir into which the mercury can flow.
Thermometers for Special Purposes.

The Clinical Thermometer. — This thermometer is intended
to read the temperature of the human body ; the range
therefore is small, usually from 95° F. to 110° F.,1 and so the

1 Fahrenheit, writing of the use of the thermometer to detect " fevers
and other distempers," observes that it is sufficient for most cases to

Milk
Glass

Section

of
Stem,

0

FIG. 10.

FIG. 11.

TEMPERATURE AND THERMOMETRY 19

stem can be short although the graduations show fifths of
degrees. It is of the thick-walled type, but the special point
is that the bore is very constricted at one point (Fig. 12).
When the thermometer is used, the expanding mercury can
force its way past the constriction, but when the thermometer
is removed from the patient the mercury contracts, and the
weight of the column above A is not sufficient to force it back
through the constriction and so it stays where it is, indicating
the temperature reached. It can thus be read at leisure.
The mercury is forced back again for further use by shaking
the thermometer sharply.

It will thus be seen that the clinical thermometer is a short
range maximum thermometer. These thermometers are
made in different sizes, known as " half minute," " one

Fio. 12.

minute," etc. This means that the quantity of mercury and
the thickness of the glass is such that the time indicated gives
some idea of the time required for the thermometer to take
up the temperature of the patient. In practice, however,
a half minute thermometer is usually allowed about one
minute, a one minute is allowed two, and so on. These
allowances are probably excessive under good conditions,
but it is obviously better to err on that side as the thermometer
should not indicate too high a temperature whatever time is
allowed.

Maximum and Minimum Thermometers. — For some pur-
poses, e.g. in meteorological work, it is necessary to know
the highest or the lowest temperature recorded during a
certain interval, say during the day or the night. Ther-
mometers capable of indicating this are known as maximum

have a thermometer reading up to 120° F., "as I have seldom come
across persons having a higher temperature than 120°." His experi-
ence must have been remarkable if he ever came across any one with a
temperature anything like this, although there has been such a case
recorded : the patient died.

20

HEAT

thermometers or minimum thermometers respectively. One
type of maximum thermometer is arranged as follows : —

A mercury thermometer is made with a horizontal stem.
A small light index, shaped like a small bar bell and usually
made of iron or glass, is placed in the tube. The index can
slide freely along the tube. When the mercury expands, the
surface of the liquid pushes the index before it. This is due
to an effect known as surface tension, which makes a liquid
behave as if the surface were formed of a sort of skin, so that
a definite effort is needed to push an object through it (an
example of this is the well-known trick of making a needle
float on the surface of water). When the mercury contracts
it leaves the index behind, so that the left hand end of the
index shows the furthest position reached by the surface of

(P)

rdSri b

1 "I I

0 10 0

1

i I I I I ' I I I I I ' I

0 20 30 40 50 60 70 I

FIG. 13.

the mercury and so indicates the maximum temperature
reached. The instrument is re-set by tilting the tube and
causing the index to slide down to the surface of the mercury.
Much the same principle is employed in the minimum
thermometer (Fig. 13). In this case the liquid used is alcohol
and the index is immersed in the liquid. When the liquid
expands it flows past the index (the ends of which do not fill
up the tube), but when the liquid contracts and the surface
reaches the index, it pushes it back rather than break the
surface. The position of the right hand end of the index,
therefore, represents the lowest temperature attained. The
instrument is re-set by tilting up the left hand end of the
tube so that the index slides down to the surface of the liquid.
These instruments are known as Rutherford's maximum
and minimum thermometer. For maximum thermometers

TEMPERATURE AND THERMOMETRY

21

P

Rutherford's form is not very much used. A very usual
form is Negretti's, which works on the same principle as the
clinical.

Six's Maximum and Minimum Thermometer. — This is a
combined maximum and minimum instrument. A (Fig. 14)
is the bulb of the thermometer,
and its tube is bent as shown and
terminates in a bulb B. A is filled
with alcohol, which extends as far
as E. The part E F is filled with
mercury. The tube above F and
part of the bulb B are filled with
alcohol ; the space above B being
occupied by alcohol vapour. The
part A E is the thermometer pro-
per, so that we are using an alcohol
thermometer. When the tempera-
ture rises the alcohol expands,
pushing the column of mercury
down in E and up in F. When
the temperature falls the alcohol
in A contracts and the pressure of
the vapour in B drives the mer-
cury column back. Scales are
placed on each limb, giving the temperature from the posi-
tion of F or E.1 Above E and F are two small light steel
indexes, provided with small springs capable of support-
ing them in the tube. When the temperature goes down,
the end E of the mercury pushes up its index, while the
other is left behind. When the temperature rises, the end
F of the mercury column pushes its index up, while the
other is left behind. Thus the lower end of the right hand
index gives the minimum temperature, and the lower
end of the left hand the maximum. The indexes are

1 Since the mercury column itself expands the right hand scale
divisions are not exactly equal to the corresponding divisions on the
other scale.

14.

22 HEAT

reset by means of a small magnet. It should be noticed that
the mercury column is there simply to move the indexes ;
the thermometric liquid is the alcohol in A. Six's ther-
mometer is not suitable for accurate work, but is in much
favour with gardeners.

The Relation between the Three Scales of Tempera-
ture.— We have seen that in each scale the two fixed points
are the same, i.e. the melting-point of ice and the boiling-
point of water under standard atmospheric pressure. In the
Fahrenheit scale the interval between the fixed points is
divided into 180° ; in the centigrade and Reaumur scales
this interval is divided into 100° and 80° respectively. Thus
a range of 180° F. = a range of 100° C. = a range of 80° R.,
or 9 Fahrenheit degrees = 5 centigrade degrees = 4 Reaumur
degrees.

Now when we wish to express a temperature as measured
on one scale, in terms of another, this relationship will enable
us to do it. But in the case of the Fahrenheit scale the lower
fixed point is called 32° not 0° ; so that the three scales do
not all start from the same point. In converting scales from
one temperature to another, the point to keep in mind is
how many degrees is the temperature above or below freezing-
point.

For example, let us express 98° F. in the centigrade scale.
98° F. = 98 — 32 or 66° above freezing-point.
9 Fahrenheit degrees = 5 centigrade degrees.
66 Fahrenheit degrees = $ x 66 = 36-6 centigrade degrees.
/. 98° F. = 36-6 centigrade degrees above freezing-point.

= 36-6°C. (since freezing-point is 0°C.).
Again, suppose we wish to exp ess a temperature of 15° C.
oil the Fahrenheit scale.

15° C. = 15 centigrade degrees above freezing-point.
_ 15 X 9.

5

= 27 Fahrenheit degrees above freezing-point.
But freezing-point = 32° F.
.-. 15° C. = 27 + 32 = 59° F.

TEMPERATURE AND THERMOMETRY 23

In dealing with conversions from centigrade to Reaumur
and the reverse it is not necessary to consider this question
of freezing-point since both scales start from the same point.
Thus, to express a temperature of 18° C. on the Reaumur
scale,

6 centigrade degrees = 4 Reaumur degrees.
18° C. = t X 18 = 144° R.

But whenever the Fahrenheit scale is involved, it is well worth
while introducing the step of the number of degrees above
freezing-point, otherwise the 32° is apt to be added or sub-
tracted at the wrong stage, and a serious error introduced.

Temperatures below freezing-point are expressed on the
centigrade and Reaumur scales by the use of the minus sign,
thus — 10° C. = 10 centigrade degrees below freezing-point.
On the Fahrenheit scale in the same way, temperatures so
low as to be below 0° F. are expressed by means of the minus
sign. The conversion from one scale to the other is precisely
the same whether the temperatures are below or above
freezing-point.

Example. — Express 10° F. as a centigrade temperature.
10° F. = 32 — 10 = 22 Fahrenheit degrees below freezing-
point.

22 x 5

= — = 12-2 centigrade degrees below freezing-
point.

= — 12-2° C.

As another example, express — 20° C. on the Fahrenheit
scale.

— 20° C. = 20 centigrade degrees below freezing-point.
_20 x 9

5

= 36 Fahrenheit degrees below freezing-point.
= 36 Fahrenheit degrees below 32° F.
= 32° - 36°.
= - 4° F.

Of the three scales, that of Fahrenheit is the oldest, having
been introduced by him about 1765 ; it was, in fact, the
first satisfactory scale used. His idea was to avoid using the

24 HEAT

minus sign, so he made a scale in which the lowest tempera-
ture he met with in his travels in Iceland should not come below
zero. The actual value of the zero he took as the lowest
temperature he could obtain with a mixture of ice and salt.
He did not, in his earlier work, use the melting-point of ice
and the boiling-point of water as fixed points and the value
of the degree seems to have been based on various considera-
tions.1 He did, however, in the end use the two fixed
points now adopted.

Note on the Definition of Temperature. — We have seen
that temperature is a condition or state of a body, which we
appreciate primarily by our senses. The test for whether two
bodies are or are not at the same temperature is given by
whether when they are placed in contact neither of them
changes its condition. A very unsound, but frequently-
encountered definition is " One body is said to be at a higher
temperature than another, if, when they are placed in contact,
heat passes from the first to the second."

This is unsound because the idea of the existence of heat
came as an attempt to picture what was happening to account
for our sensation of hotness. It is probable that there is
something corresponding to our conception of heat (as we
shall see in Chapter VIII heat is a form of energy), but it is
not directly perceptible to us ; to introduce it in the definition
of temperature is arguing in a circle or is meaningless. A more
detailed consideration of scales of temperature will be met
with later (see Chapter XVI, p. 253).

1 One of his earlier scales made use of the human body temperature
as a fixed point. In one of his later papers he mentions that a mixture
of alcohol and water in the proportion of 4 : 1 expands ^os of its
volume at freezing point (of water) for each 1° of his scale and seems to
imply that he made use of this fact in fixing his degrees. It is worth
noticing that mercury expands almost exactly TrF<7or> of its volume at
32° F. for each 1° F. rise in temperature, although this point is not
mentioned in his papers.

CHAPTER m
EXPANSION OF SOLIDS

We have seen that most solids expand when heated ; this
expansion is of great importance in practical life. In some
cases it can be made to serve useful purposes, while in others
it is a nuisance for which provision has to be made. There
are two important points to be noticed in connection with
this expansion. In the first place the force exerted during
the expansion of a solid when heated, or during its contraction
when cooled, is very great, and secondly it is found that
different solids expand by different amounts when heated
through the same range of temperature.

The great force exerted during expansion or contraction
may be illustrated by means of the apparatus known as the
" bar breaker." It consists of a stout iron rod A (Fig. 15),

FIG.

having near one end a hole large enough to admit easily a
short bar of cast iron about f-inch diameter. On the other
end of the rod is cut a strong screw thread on which a large
nut works.

The rod is placed in a massive iron stand, so that the bar
F rests against the stops BB. The faces of these stops are
V-shaped so that the bar presses against two edges instead

25

26 HEAT

of two flat surfaces, and the edge of the hole in A is
V-shaped. The nut is turned as far as it will go, so that it
presses against the stops CO. Thus A is prevented from
expanding by the pressure of the bar F against the stops at
one end, and the nut against the stops at the other. The
gas burner G (a tube closed at one end and provided with a
number of slits) is lighted, and as the rod gets hot the force
it exerts is sufficient to snap the cast-iron bar.

The experiment may be varied so as to use the force of
contraction. In this case the nut is screwed up against the
stops EE, the cast-iron bar resting against DD. The rod is
heated and as it expands the nut is screwed up against EE
so as to take up the expansion. The rod is then allowed to
cool, and the force it exerts in trying to contract to its former
length breaks the bar as before.

The fact that different solids expand by different amounts
under the same conditions may be illustrated in the following
way. A strip of copper and of iron are securely riveted
together. The two strips should be fairly thin and the rivets
placed at intervals of about half an inch along each side.
When the compound bar so formed is heated by a burner
the copper expands more than the iron, and since it cannot
slide over the iron without tearing away from the rivets it
makes room for itself by bending the bar into a curve, the
iron being on the inside and the copper on the outside. The
bar is therefore seen to bend in this way when heated and to
straighten again in cooling.

One of the most obvious cases in which expansion becomes
a nuisance is the measurement of lengths by measuring rods
or scales. The rod can only be correct at one particular
temperature, and errors will be introduced unless all the
measurements are taken when the temperature is at the
particular value. This change in length of the measuring
rods was a source of great difficulty in the earlier surveys of
Great Britain, and it was in connection with them that much
of the first systematic study of expansion was made.

In surveying a large area the process of triangulation is

EXPANSION OF SOLIDS 27

adopted. In this process two points A and B are selected,
and the distance between them carefully measured. This
length AB is called the Base Line. A third point C visible
both from A and from B is selected, and the angles CAB and
CBA are measured by means of a theodolite. Knowing
these two angles, and the length AB, the lengths CA and CB
can be calculated. These are then used as the bases of two
new triangles and the distance of two more points calculated.
In this way the whole area is covered by a system of triangles,
and it will be seen that the only length actually measured is
that of the Base Line. Upon this measurement depends the
accuracy of the whole survey.

In practice it is found desirable to have a base line of con-
siderable length such as five or six miles, and as the accurate
measurement of such a length is an affair of months, consider-
able variations in temperature are likely to take place while
it is being carried out. Unless, therefore, the changes in
length of the measuring rods have been studied in detail
accurate surveying is impossible. For this reason, when
General Roy made a survey of Hounslow Heath in 1784,
upon which was founded the first Ordnance Survey of England,
he had an apparatus designed for him by Ramsden and spent
a considerable time investigating the behaviour of his measur-
ing rods. Lavoisier and Laplace had made some very careful
experiments on the same subject a few years before, but the
account of them was not published until nearly twenty years
later.1 It is also worth mentioning that as early as 1754
John Smeaton had made a systematic study of the same
subject, and although his experimental work cannot be
considered as equal to these later experiments, it is valuable
as the first really clear statement of the problem to be attacked
and the difficulties to be overcome.

It was found that as far as could be determined with the
apparatus used, if a given rod increased in length by a certain
amount, say 1 mm. when heated from 10° C. to 20° C. as

1 The French Revolution was not a time which was conducive to the
pursuit and publication of scientific work.

28 HEAT

measured by a mercury thermometer, it would expand a
further 1 mm. when heated from 20° C. to 30° C., and so on.
More accurate work has since shown that this is not strictly
true for any substance, especially over a large range of
temperature : but in most cases it is so nearly true that for
moderate ranges of temperature it can be taken to be true
for the great majority of practical purposes. To a fair degree
of accuracy, therefore, we can say that a given rod increases
in length by a certain amount for each degree rise in the
temperature : the amount depends on the material of which
the rod is made and obviously also on its original length.

If, instead of speaking of the rod expanding a certain
amount per degree we say that it expands a certain fraction
of its original length per degree, this fraction will depend only
on the material. This, in practice, is done ; since, however,
the original length of the bar depends on the temperature at
which it is measured some standard temperature must be
agreed upon. It is usual to take the " original length " as
the length at the temperature of melting ice. The fraction
is called the Coefficient of linear expansion of the material.
It is therefore defined as follows : —

Definition of Coefficient of Linear Expansion. — The

coefficient of linear expansion of a substance is the fraction
of its original length (measured at the temperature of melting
ice) by which a bar of the substance would increase in length
for each degree through which its temperature is raised.

Thus when we say that the coefficient of linear expansion
of iron is -000012 we mean that a bar of iron 1 cm. long at
0° C. would expand -000012 cm. per degree ; a bar 1 foot
long at 0° C. would expand O00012 feet per degree ; a bar
6 feet long at 0° C. would expand 5 X -00001 2 feet per degree,
and so on.

It may be noticed that the bar also increases in thickness,
but since we are concerned only with linear expansion this
does not affect the result ; if we were concerned with the
total change in volume we should speak of cubical or volume
expansion instead of linear (see Chapters V and VI).

EXPANSION OF SOLIDS

29

Measurement of the Coefficient of Linear Expansion.

We shall consider two laboratory methods of measurement.
The first is rather rough and only suitable for giving approxi-
mate results ; it is, however, quite satisfactory as far as it
goes. The second is more accurate, being intended to give
results trustworthy to 1 per cent. ; it will be instructive to
notice the much greater care and trouble involved in accurate
work.

Method I. For Approximate Results. — This method
is only suitable for metals. A tube of the given metal, about
60 cm. long and 0-5 cm. in diameter, has a mark made near
each end. The distance between these marks is measured
by means of an ordinary metre scale. The tube is clamped
at one end, and at the other rests on a small roller carried by an
arm attached to a rigid stand. The clamp is placed over one
mark, while the roller touches the tube at the other. This
roller is of known diameter, (usually 2 cms.), and to it is
attached a pointer about 20 cm. long. A millimetre scale,
fastened to a small block of wood, is placed on the bench
just behind the pointer. The tube is connected by means

FIG. Ifl.

30 HEAT

of a length of rubber tubing to a boiler from which steam can
be supplied.

The metal tube CD is first allowed to take up the tempera-
ture of the room, which is determined by means of a ther-
mometer, and the reading of the end of the pointer A noted :
it is desirable to adjust the position of the scale so that the
first position of the end of the pointer is exactly opposite a
scale division.1

Steam is then passed through the tube ; as the latter
expands the pointer moves over the scale. When no further
movement takes place and the pointer has been at rest for
two or three minutes, the reading of its extreme end is again
noted. The tube is now removed, and the length of the
pointer measured.

The following readings were obtained in a certain experi-
ment and will be used to illustrate the method of obtaining
the coefficient : it will be observed that the experiment
happened to be done in a cold room.

Brass Tube.

Temperature of the air . . . „ 9° C.

Temperature of the steam .... 100° C.
First reading of end of pointer , . * 5-0 cm.

Second 3-3 cm.

Diameter of roller . . . . , .2-0 cm.

Length of pointer . . . . . .19-0 cm.

Distance between marks on tube . . .50-0 cm.

Since the tube is clamped at C, the movement of the mark
D represents the total expansion of the length of tube between
the two marks ; the expansion of the two overhanging
portions obviously does not- affect the roller. The movement
of the tube causes the roller to turn 2 and with it the pointer.
The pointer is 19 cm. long and the roller is 1 cm. in radius ;

1 The scale should be as close to the pointer as possible, without
actually touching. See note on parallax error.

2 Care must be taken that the pressure of the tube on the roller is
sufficient to prevent any chance of the latter slipping back.

EXPANSION OF SOLIDS 31

hence the end of the pointer moves in the arc of a circle of
20 cm. radius, i.e. 20 times the radius of the roller. Thus
the end of the pointer moves 20 times as far as a point on the
circumference of the roller.

Now the end of the pointer moved through a distance of
1-7 cm. (It is true that the distance was measured along
a straight line, while the end moved in the arc of a circle ;
for such a small arc, however, we can neglect the error as
explained on page 37).

Consequently the point moved through — = -085 cm.
The rise in temperature was (100 — 9) = 91°.

Hence the expansion per degree = -— = -00093 cm.

91

Therefore the coefficient of linear expansion

•00093 cm.
length CD at 0° C.

To obtain the length CD at 0° C. we make use of our deter-
mination of the expansion per degree. The length of CD
was measured at 9° C. ; if we cooled the tube to 0° C. we see
from our results that we should diminish the length by
9 X -00093 cm. = -0084 cm. or less than -1 mm. But we
measured CD to the nearest mm., and hence should not have
been able to detect -1 mm. Hence for this work we can regard
50-0 cm. as sufficiently near the length at 0° C. for our
purpose.

The coefficient of linear expansion of brass is therefore

•00093 cm. = .QooQig (to two significant figures).
50'0 cm.

Method II. More Accurate Method Intended to Give
Results Trustworthy to about 1 per cent. — The apparatus
was designed to measure the expansion of metal tubes so that
the material used in the first method could be used.1

The arrangements of the previous experiment are modified
in three ways.

* For the case of solid rods, see note, page 35.

32

HEAT

(1) A more accurate instrument is used for measuring the
expansion.

(2) More careful arrangements are made for ensuring that
the tube is really at the temperature which it is supposed to
be.

(3) Provision is made that the increase in length shall not
be obscured by any chance expansion of the measuring
apparatus.

The tube is lagged with a thick covering of cotton wool l
for the whole of its length with the exception of two gaps
about 1 cm. long over each of the two marks. Another
similar tube also lagged, through which cold water can be
passed, is placed parallel to it ; this tube is to serve as a constant
standard of length. The two tubes are clamped near one end
in the special form of clamp illustrated ; the two edges B B

FIG. 17.

(Fig. 17) are of hard wood and are about 3 mm. thick, the
remainder of the clamp being iron. The experimental tube
is clamped at one of the marks and the portion near the other
mark rests in a V-notch in a narrow upright piece of wood
screwed to a block. The tube is thus free to expand in this
direction. Two fine marks, about 1 mm. apart, are made
close to the larger mark and are viewed by means of a micro-
scope fitted with an eyepiece micrometer scale (the ordinary

1 For convenience in putting on, the cotton wool is wound on a
thin cardboard tube which just slips over the brass tube.

EXPANSION OF SOLIDS 33

mark when viewed through the microscope presents too coarse
an outline for accurate measurement and two fine ones are
made so that one may be used as a check on the other).

Clamped to the cold water tube is an arm terminating in a
fine needle, the point of which is adjusted to be as near as
possible to the marks on the experimental tube, so that it is
seen through the microscopes at the same time as the marks
and serves as a standard of reference.1 The cold water tube
also rests in a notch in a piece of wood placed to one side of
the other to allow room for the clamp. Lengths of rubber
tubing are connected to each end of the two tubes. Cold
water from the tap comes in at C. D is fitted with a piece of
glass tubing, which is at first connected to E, so that water
circulates through both tubes and passes out at F.2

The water is allowed to circulate for some time (e.g. 15
minutes), until the temperature is constant ; this temperature
is noted. The position of the pointer and of the fine marks
is then read off on the eyepiece scale. E is then uncoupled
from D and connected to a boiler so that steam passes through
the experimental tube, while cold water continues to pass
through the other. The reading of the two marks and the
pointer is again noted ; it will be found that the marks move
through a considerable distance almost immediately, but do
not attain their final position until the steam has been passing
for some time, e.g. 10 or 20 minutes. Headings are taken
every few minutes until it is quite clear that they have become
constant. E is then uncoupled from the boiler and connected
to D so that cold water circulates through both tubes ; when
everything is steady readings are again taken to see that the
tube returns to its original length. As a general rule the
reading of the pointer remains the same throughout as the
bench does not become heated to any great extent ; sometimes,

1 It is difficult to get the pointer in focus in the field of view and at
the same time not touching the tube ; the writer finds that the most
satisfactory arrangement is to use a small electric lamp to illuminate
the marks and then one can sight on the shadow of the pointer.

1 The final outlet should be above the level of the metal tubes to make
sure that the latter are always full.

D

34 HEAT

however, there is a shift of -01 mm. or so, which usually means
that all the final readings are different from the initial ones —
the bench does not return very quickly to its former condition,
or there may have been some slight shift of the microscope —
and it is necessary to have the pointer as a check. The
pointer also serves as a useful reminder if the observer happens
to lean on the bench when taking a reading and so causes a
distinct shift. Either before or after the experiment the
eyepiece scale is calibrated by focussing the microscope on a
millimetre scale and counting the number of divisions which
correspond to a whole number of millimetres.

The following figures from an experiment illustrate the
method.

Temp, of cold water = 8° C. Temp, of steam (bar. = 746
mm.) = 99-5° C.

Readings 123

Pointer . . .25-0 25-0 25-0

1st mark . . . 28-2 20-8 28-2

2nd mark . . . 33-8 264 33-8

Movement of each mark = 7-4 scale divisions.
Calibration. 3 mm. corresponds to 25-8 scale divisions.
1 scale division = -1163 mm.

Expansion of the tube = 7-4 x -116 = -86 mm.
Expansion per degree = -86/91 -5 = -094 mm.
Coefficient = -094/500 = -0000188.

Note on the Experiment. — It will be seen that in order to
get the value of the coefficient correct to 1 per cent, the
expansion must be measured to an accuracy well within -01 mm.
Travelling microscopes worked by a micrometer screw are
made, which read to -01 or even -001 mm., but unless they
are in good condition and are of first rate design and make,
a number of determinations of the distance between two close
marks will differ by considerably more than -01 mm. On
the other hand it is not difficult to obtain an eyepiece scale
which with a very ordinary microscope will read consistently
to -01 mm. if the focussing is accurate and the illumination
good. An eyepiece scale is therefore recommended, and it
ia also less trouble.

EXPANSION OF SOLIDS 35

Where the specimen is in the form of a rod and not a
tube it must be surrounded by a jacket and project a very
small amount from each end (readings taken through a glass
jacket are very unreliable unless they are taken through flat
windows of optical glass). Supporting the rod by its extreme
ends is likely to produce errors due to sagging, and it is better
to support it at two points within the jacket (at positions of
minimum sagging given by Airey 's formula1) and use a micro-
scope at each end with a constant temperature standard of
length. In this case the arrangement becomes a sort of
comparator (page 39) ; laboratory comparators are made e.g.
by the Cambridge and Paul Instrument Company, but they
are very expensive, and in few laboratories is a really accurate
value for a coefficient required sufficiently often to justify
the expense.

Note on the Degree of Accuracy of Physical Measurements.

No practical measurement is absolutely exact ; it can only
be approximate. On the other hand, if the measurement is of
use it is most important that the degree of accuracy should be
known. Suppose, for example, that the length of a rod is being
determined by means of a scale graduated in mm. The scale
would be placed so that one end of the rod is as nearly as can be
judged opposite to a division, and the observer would then be
able to say which division of the scale was nearest to the other
end. Thus if the length were recorded as 87 mm. it means that
the observer is satisfied that 87 is nearer the length than 88 mm.
or 86. In other words, he is satisfied that the length is some-
thing less than 87-5 mm. and greater than 86-5 nun.

This might be written 87J^O-5 mm., where 0-5 mm. is called
the " probable error." With accurate instruments and skilled
observers the probable error of any measurement can be made
very small, but it cannot be eliminated altogether.

When a result is based on several observations it is easy to
be misled as to its accuracy. For example, suppose that the
area of a rectangular surface is being calculated. The length
and breadth are measured as above to the nearest mm. and

1 Cp. Travaux et Memoires du Bureau International dee Poids et
Mesurei. Vol. II.

36 HEAT

found to be, say, 9-4 cm. and 8-6 cm. respectively. At first
sight it might be thought that it is legitimate to say

Area of rectangle =9-4 x 8-6 = 80-64 sq. cm. or for safety's
sake 80-6 sq. cm.

But the length is really only known to be between 9-35 and
9-45 cm. and the breadth between 8-55 and 8-65 cm. Now
9-35 x 8-55 = 79-9425 arid 945 x 8-65 = 81-7425. Thus we
can only say with certainty that the area is between 79-9425 sq.
cm. and 81-7425 sq. cm.

Thus the area should be written not as 80-64 sq. cm., but
81 sq. cm. and too much reliance should not be placed on tho
last figure ; the area might be nearer 80 or 82 sq. cm. In fact
the result should only be worked out to as many significant
figures as the least trustworthy measurement, and even then the
last figure is not very reliable.

If the area were recorded as 80-64 sq. cm., not only has time
been wasted in useless multiplying, but what is far worse an
appearance of accuracy has been given which the measurements
do not warrant ; it looks as if it might be trusted at least to the
first decimal place whereas it is really hardly reliable to the
nearest whole number.

It is instructive to examine the results of the experiment
described on page 29 from this point of view. The final result
depends on three sets of observations, i.e. : —

(a) The increase in length of a portion of the brass tube.

(6) The original length of this portion.

(c) The range of temperature through which the tube is heated.

(a) In this reading we have to consider the question of the
probable error in reading a mm. scale. The position of a point
actually touching the scale might fairly easily be read to the
nearest half millimetre, i.e. a good observer could tell whether it
was nearer to a division than to a point halfway between the

divisions — a probable error of +-25

; 1 mm. But the pointer must not

touch the scale lest it should be

occue / slightly bent or the scale moved, and

S an error due to parallax may be in-

troduced owing to the eye not being
exactly opposite to the point er when

Eye taking a reading. Let Fig. 1 8 repre-

FlQ, 18. sent a plan of the pointer and scale.

EXPANSION OF SOLIDS 37

The true reading of the pointer is A, i.e. A is directly behind
P. But to an eye placed as shown, the reading of the pointer
would appear to be at B. The distance AB is the parallax error.
The figure of course represents a very exaggerated case, but
some error may be introduced. In view of this danger, and also
the fact that a very slight want of rigidity in the stands of the
clamp and roller might cause a movement of the tube and con-
sequently a greater movement of the pointer we ought not to
expect to get a reading for the movement of the pointer with a
probable error of less than *5 mm. Hence our reading of 1-7
cm. means anything between 1-65 and 1-75 cm.

The correction for the fact that the end of the pointer moves
in the arc of a circle while the distance is measured along a straight
line is obtained thus : —

The angle through which the pointer has turned is about

1*7

tan"1 — - or a little less than 5°. The difference between the cir-
cular measure and the tangent for such an angle is about 1 in 800
(for 14° it is about 1 per cent, and increases rapidly as the angle
gets greater).

Hence our correction would be about O^T: cm. = -002 cm.

As we have already a probable error of -05 cm. we need not
trouble about a further -002 cm.

(6) The length of the tube was measured by a mm. scale and
found to be 50*0 cm. An error of half a centimetre would only
be 1 per cent, of the length ; hence our scale is more than suffi-
ciently accurate as a means of measurement, and what is more
important, a slight uncertainty as to the exact point held by the
clamp does not make this reading less reliable than the others.

(c) Temperature can be determined by means of a fairly good
thermometer within about a degree : the temperature of the steam
is unlikely to differ from 100° C. by as much as one degree, as
this would involve a barometer reading above 787 mm. or below
733. On the other hand, when the steam is passing through the
tube, the inside of the latter may take up the temperature of the
steam, but the outside is in contact with the air and will be at a
lower temperature. In the case of a good conductor like brass
the error in estimating the temperature of the tube might be one
or two degrees ; for a poor conductor it would probably be more.

38 HEAT

Thus the range of temperature which we called 91° is not likely
to be more, but might easily be two degrees less.

Thus the increase in length per degree might be as high as

2Q x 89 or as low as 2Q x 91> i.e. anything between -000985 cm.

and -000905 cm. Dividing this by 50 cm. the coefficient comes out
as between -0000181 and -0000197. The result given -000019 is
therefore barely reliable to the second significant figure ; it is,
however, more likely to come out too low than too high, so that
in any doubtful case the higher value should be taken. It will
be seen that the result is likely to be within 5 per cent, of the true
value and to this degree of accuracy the method is good enough
in the case of metals.

It should be noted also that the result cannot be quite as trust-
worthy as the least accurate measurement, for even if all the
others were absolutely correct, the probable error of this measure-
ment is multiplied by the same factor as the reading itself. Hence
it is no use taking elaborate precautions to ensure accuracy for
some measurements unless the others are correspondingly precise.
It is instructive to consider the second method in the above
manner, and trace out how the accuracy of 1 per cent, is justified.

The Standardization of Measuring Rods.— The standard-
ization of measuring rods is one of the most important cases
where an accurate knowledge of coefficients of expansion is
required : it was in connection with this need that the first
systematic work was done. Modern methods may be regarded
as the development of the work of Roy and Ramsden,1 influenced
also by some aspects of that of Lavoisier and Laplace.

The chief points in Roy's method were : (1) the whole appar-
atus carrying the measuring instruments was maintained at a
constant temperature throughout, being surrounded by melting
ice ; (2) the increase in length was measured directly by a micro-
meter combined with an optical arrangement of great accuracy.
The chief, if not the only, defect from a mechanical point of view
arose from the fact that the measuring instruments were carried
on uprights and any want of rigidity in the apparatus would
introduce errors. Lavoisier and Laplace f had adopted a different

1 Phil. Trans. 1785. The account is well worth reading.

2 Experiments carried out in 1781 and 1782 ; the papers published
in 1803 contain letters implying the existence of diagrams. J. B. Biot
in his Traite de Physique (1817) drew figures from the description and

EXPANSION OF SOLIDS 39

plan : they arranged that the expansion of the rod should be
magnified by a system of levers including a tilting telescope, so
that in the final experiments an apparent movement of 744 scale
divisions corresponded to an expansion of yV of a Paris inch. It
was difficult to be sure of the exact amount of the magnification
and the measurement was less reliable than Roy and Ramsden's
direct method.

The chief interest of this method is that in order to obtain
rigidity the supports of the lever system were carried on stone
blocks (2 feet square by 1 foot thick) on masonry 6 feet deep
(the experiments were carried out in the Tuileries gardens) ;
moreover Lavoisier and Laplace pointed out that the most
difficult measurement, and that most liable to error, was the
temperature determination. They accordingly used thermo-
meters graduated to tenths of a degree, which were compared
to a standard.

The modern development of these two pieces of apparatus is
the " comparator " designed and used by the International
Committee of Weights and Measures at Paris.1 The comparator
is used for comparing the length of any metre scale, heated in
succession to different temperatures, with that of a standard
metre kept at constant temperature.

Attached to two massive stone pillars are microscopes A A
(Fig. 19) as nearly as possible 1 metre apart, and each fitted
with a micrometer. The standard metre is placed in one trough,
and the scale to be compared with it in another, parallel to the
first. The two troughs are carried on a kind of table resting on
wheels, which run on three rails, so that first one trough and then
the other can be brought under the microscopes. As the table
and troughs are very heavy, the rails are carried on a massive bed
with a concrete foundation.

The troughs are filled with water, which in the trough contain-
ing the standard is kept at a constant temperature, while in the
other it is heated to different temperatures in turn. In order to
do this the troughs are double walled, the scale being immersed
in water in the inner trough, while in the space between the walls
circulates water from a reservoir kept at a constant temperature

from his memory. Two sketches were found by the Editor of Lavoi-
sier's works and were published in 1893.

1 Travaux et Memoires du Bureau International des Poids et Mesure*.
Vol. II.

40

HEAT

by means of a thermostat, i.e. an arrangement where the tem-
perature of the water in the reservoir regulates the supply of gas
to the burners which are used in heating it. By altering the
adjustment of the thermostat any desired temperature can be
maintained.

Very efficient stirring apparatus is fitted to the troughs, and the

FIG. 19

temperature of the water in the inner trough is given with great
accuracy by thermometers which have been carefully studied.

The standard metre is first brought under the microscopes and
the micrometers adjusted until their cross wires are over the end
marks of the scale ; the other trough is then brought under the
microscopes and the micrometers again adjusted until the cross
wires come over the ends of this scale. The total movement of
the two micrometers gives the difference in length between the
standard and the other scale. The standard is then again brought
under the microscopes, and the micrometers again adjusted. If
the readings are not the same as at first it means that there has
been some expansion of the supports due to the heating effect of
the other trough, and a correction has to be made.

CHAPTER IT
PRACTICAL CONSEQUENCES OF EXPANSION

In addition to the case of measuring rods other cases arise
where expansion has to be considered. A familiar example is
that of railway lines. If the rails were all in one piece their
length on a hot day would be considerably greater than on a
cold. Take, for example, the line from London to Edinburgh
— roughly about 400 miles. Taking the coefficient of linear
expansion of iron as -000011 this length of rail would expand
by -000011 X 400 miles for each 1° C. rise in temperature.
A difference of 25° C. is quite an ordinary difference between
a cold day in winter and a warm day in summer ; this repre-
sents an expansion of 25 X 400 X -000011 miles, or about
194 yards. As the force of expansion is very great, the rails
would probably buckle. In consequence a small gap is left
between each length of rail and the next to allow for this
expansion and contraction.

Below the surface of the earth the variations of temperature
between winter and summer are much smaller, and at a depth
of a few feet become negligible. At such a depth the tempera-
ture is practically constant all the year round. Consequently
the rails of tube railways do not require to have gaps between
each length, and can in fact be welded together. If, however,
the railway comes above ground, gaps have to be left. An
example is to be found on the G.N. Piccadilly and Brompton
tube ; the section between Hammersmith and Baron's Court
is above ground and gaps are left between each length of rail,
and also between each length of the conductors, from which
the motors draw their supply of electricity. As soon as the
line gets below ground, these gaps are no longer left. Even

41

42 HEAT

in the case of tramway lines, which are merely embedded a
few inches in the ground, it is found that gaps are unnecessary
and the rails can be welded together.

Many other examples of expansion are to be found ; water
mains exposed to the variations of temperature in air have
to be fitted with telescopic joints, and in the same way the
tubular girders of the railway bridge across the Menai Straits
are mounted on rollers and have a play of several inches.
Telegraphic wires can be observed to sag more in hot weather
than in cold, and if the wire be put up during hot weather
sufficient sag must be allowed for it to have room to contract
in cold weather.

The Behaviour of Glass. — Glass vessels which are in-
tended for heating, such as flasks, beakers, etc., must have very
thin walls ; thick glass would crack. This is because glass
is a bad conductor of heat, so that when the outside of the
vessel is heated it takes some time for the glass to be warmed
throughout. The outside therefore expands, while the inside
is still cool and has not expanded, and so the glass is strained
and cracks. If the glass is so thin that it quickly gets heated
throughout, it can safely be heated as long as the heating is
not too sudden or too local.

Since the introduction of the electric furnace it has been
possible to make vessels of fused silica, which is not unlike
glass in appearance but very brittle and very infusible. Quite
thick vessels of this substance can be heated to redness and
suddenly plunged into cold water without injury. Silica is
a bad conductor like glass, so that there would be great
difference in temperature between the inside and outside,
but the coefficient of expansion is only about TV of that of
glass and so the strains are quite small.

Glass of any thickness, which has been heated and allowed
to cool quickly, is in a state of great strain due to the unequal
contraction. The two scientific toys known as Prince
Rupert's drops and Bologna flasks, are well-known examples
of this. The former are lumps of glass formed by heating a
glass rod and allowing the molten glass to fall in drops into

CONSEQUENCES OF EXPANSION 43

cold water. The drops form pear-shaped lumps, with a sort
of tail, and are in a state of great strain. If the end of the
tail be broken off the whole drop flies to pieces with a loud
report.

Bologna flasks are simply bulbs blown on the end of a short
piece of thick glass tubing and suddenly cooled by plunging
the softened glass into an iron mould. If a very small piece
of flint be shaken about inside it will make quite small scratches
(usually invisible) and cut through the inner layer of glass
and disturb the equilibrium. This causes the bottom of the
flask suddenly to crack and drop out.

Cheap glass tumblers and dishes sometimes emulate these
toys and suddenly crack as the result of a slight knock or the
pouring in of hot water. Better quality glass vessels are
carefully " annealed " after being made ; that is, they are
heated almost to the softening point in a muffle, maintained
at this temperature for some time so as to get warmed through,
and then allowed to cool very gradually. The glass thus
contracts evenly and strains are avoided. Such vessels are
much less likely to crack. This point must be remembered
in all glass blowing work in the laboratory ; a sealed
joint is almost certain to break if it is allowed to cool
quickly.

In many pieces of apparatus it is necessary to have wires
passing through the sides of glass vessels, the joint being air-
tight. To make a successful sealed joint the wire must have
the same coefficient of expansion as the glass (since the latter
has to be heated to the softening point). Platinum has
practically the same coefficient as glass and is therefore the
metal largely used (but see p. 49).

Where it is essential to have a wire of different material,
much trouble is involved. Thus in making the Coolidge
X-ray tube it was necessary to have molybdenum wires at
one point. A special glass having a coefficient equal to that
of molybdenum was made and the wires passed through a
piece of this glass ; this glass was then connected to the rest
of the tube through a series of intermediate glasses whose

44 HEAT

coefficients of expansion differed by such small steps that they
could be sealed to one another.

Applications of the Principle of Expansion. — The
enormous force exerted during expansion and contraction is
frequently made use of in the process known as " shrinking."

For example, in making a cart wheel, the iron tyre is made
so that it is a little too small to slip on the wheel. It is made
red hot, when it expands, so that it can be slipped over the
wheel, which is lying down on the ground. Buckets of water
are then poured over it, when it cools and contracts, binding
the wheel together, and very firmly secured. An engineer
would refer to this as " shrinking " on the tyre, and the
process is frequently employed.

For the same reason the rivets used in boilers are put in
red hot, so that as they cool they bind the plates closely
together. This would not by itself make a sufficiently steam-
tight joint for modern high pressures ; the edges of the plates
are either " caulked " or " fullered " (caulking consists in
treating the edges with an instrument like a blunt chisel
after they have been rivetted) .

When this has been done the fact that the rivets have been
put in red hot prevents the plates becoming loose again when
the boiler is under steam.

Various types of automatic fire alarms are based on the
principle of expansion. A common type makes use of a
compound bar of brass and iron, similar to the one described
at the beginning of this chapter. It is arranged so that when
it becomes hot it bends, and touches the point of a screw, thus
completing an electric circuit and ringing a bell. A similar
device has been used for indicating whether the rear lamp of
a motor car is alight ; the compound bar being placed over
the lamp, so that as long as the lamp is alight the heat is
sufficient to keep the bar bent away from a screw point. If
the lamp goes out the bar straightens and touches the screw
point and completes an electric circuit and lights a small lamp
in front of the driver.

The principle of expansion is applied in the construction of

CONSEQUENCES OF EXPANSION

45

thermostats. These are arrangements for maintaining a
constant temperature. Fig. 20 shows a simple laboratory
pattern suitable for a water bath. The thermostat is placed
in the bath and the gas supply passes through it to the burner
which is doing the heating. The gas comes in by the glass
tube D and travels as shown by the arrows to E and so to the
burner. The bulb A contains toluene
or some similar liquid with a large
coefficient of expansion ; the lower
part of A and the narrow tube con-
tain mercury. When the temperature
of the bath rises above a certain value
the expansion of the toluene causes the
mercury to close B and cut off the gas
supply ; a small by-pass at C allows
sufficient gas to pass to save the burner
from being completely extinguished.
The adjustment of the precise temper-
ature to be maintained is accomplished
by altering the position of the narrow
tube B.

Devices for Compensating the
Effect of Expansion. — The fact that
different substances have different
coefficients of expansion has been
utilized in various compensating de-
vices. The earliest examples are to be
found in the case of clocks. The
rate of a clock is governed by the
time of swing of its pendulum, and this depends on the
length (or to be accurate, on the distance between the centre
of suspension and a certain point called the centre of oscilla-
tion ; this last point will not be very far from the centre
of gravity of the " bob " if the latter be heavy and the
pendulum rod light).

The time of swing of a long pendulum is greater than that
of a short one, and consequently when the pendulum of a

FIG. 20.

4fl HEAT

clock expands on a warm day the clock would go more slowly ;
thus it would tend to lose in warm weather and gain in cold.
Compensating pendulums were designed independently
by two clockmakers named Graham and Harrison about
1720, Graham's pendulum being a year or two the earlier.
Fig. 21 shows Graham's pendulum. The rod is of iron,
n while the bob is formed by two glass tubes con-

taining mercury.

When therefore a rise in temperature causes
the iron rod to expand, increasing the length
of the pendulum, the mercury columns expand,
raising the centre of gravity of the bob, and
so diminishing the effective length of the pendu-
lum. Mercury expands very much more than
iron, so that the expansion of quite a short
column of mercury will raise the centre of
gravity sufficiently to counteract the expansion
of the iron.

The exact amount of mercury to produce
compensation was determined by trial for
each clock ; the length of the columns has to
be about one-sixth that of the rod. It should
FlG' be noted that the glass tubes increase in dia-

meter, but the expansion of the mercury is much greater
than that of the glass.

Harrison's " Gridiron " pendulum is more readily under-
stood by considering a modern variation. In Fig. 22 A is a
crosspiece firmly attached to two steel rods I, I. These carry
at their lower ends crosspieces B, to which are attached zinc
rods Z, Z. These are connected at the top by a crosspiece to
which is attached a steel rod S, carrying the bob of the
pendulum.

The expansion of the steel rods carries the bob downwards
through a distance equal to the expansion of S plus the
expansion of one of the rods I (since S itself is supported by
I, I and would be carried down through a distance equal to
the expansion of either). The expansion of the zinc rods

CONSEQUENCES OF EXPANSION 47

carries the bob up through a distance equal to the expansion
of either rod.

Now since the coefficient of expansion of zinc is about two
and a half times that of steel, it follows that by making the
length of I and S about two and a half times that of Z the expan-

I-

-s

-

-I

—c

£

B
—D

FIG. 22.

FIG. 23.

sionwill just balance and the effective length remain constant.
In the original pendulum Harrison used brass and steel,
and since the coefficient of the expansion of brass is only
about one and a half times that of steel the combined length
of the steel bars must be one and a half times that of the brass.
Since this clearly could not be done with the two pairs of rod*

48 HEAT

used in the more modern form, he used five rods of steel and
four of brass as shown in Fig. 23.

It will be clear that the total length of A, C, E and the small
piece from S to the crosspiece must be one and a half times
the total length of B and D. Harrison made a great many
experiments before he obtained a satisfactory result, since
the coefficients of expansion of brass and steel were not known
with any degree of accuracy at that time (1725).

In actual practice the pendulums have to be over com-
pensated, because the rate of a clock is affected by change
of temperature in other ways than by alteration in the length
of the pendulum, and these other effects are corrected by this
over compensation.

In the case of chronometers and watches the rate is con-
trolled by the oscillation of the balance wheel under the
influence of a spring (the "hair spring"). The time of
oscillation depends on the stiffness of the spring and on the
mass and shape of the wheel. An increase in diameter of
the wheel (the mass remaining the same) would cause it to
oscillate more slowly (just as a flywheel of larger diameter
is more difficult to start or stop), while an increase in stiffness
in the spring would make the wheel oscillate more quickly.
A rise in temperature would increase the diameter of the
wheel, and diminish the stiffness of the
spring, both of these tending to increase
the time of oscillation and so make the
watch lose. To counteract this the rim
is made in two (sometimes three or four)
sections, each section being fixed to a
spoke. The sections are compound bars
of iron and brass, with the brass on the
outside, and carry weights on their ex-
tremities (see Fig. 24).

A rise in temperature causes these sections to curl inwards,
carrying the weights with them, so diminishing the effective
diameter and compensating for the expansion of the spokes
and the diminished stiffness of the spring. This device is

CONSEQUENCES OF EXPANSION 49

due originally to Harrison, although various improvements
have been made since his time.

Compensated Measuring Rods. — It will be obvious that
a measuring rod might be constructed on the same principle
as Harrison's pendulum. It would, however, be somewhat
inconvenient and unnecessarily complicated. A simpler
form was designed by General Colby for the Indian Survey
and has been used since.

In the construction of machinery the different parts are
likely to be made at different times and under different
conditions. Since the dimensions of the parts often have to be
accurate to TUVn inch, or in some cases 1 G^G inch, or even less,
it is clear that a part made to size on a warm day would not fit
in with a part made on a cold one. To get over this difficulty
the gauges used in testing the parts should be made of the
same metal as the parts themselves ; the expansion of the
gauge will then just balance the expansion of the part, and
when they are assembled they will fit, in spite of being made
at different temperatures.

The Properties of Nickel Steel. — In the course of an
extended research on the properties of nickel steel, M.
Guillaume discovered that the coefficient of expansion is
very much influenced by the percentage of nickel. Two
particular alloys are of importance, that containing 36 per
cent, of nickel and that containing 45 per cent, of nickel.

The former has a much smaller coefficient of expansion
than alloys containing a greater or less proportion of nickel ;
to this alloy M. Guillaume gave the name of Invar. The
coefficient of linear expansion of a specimen of invar depends
on the conditions under which the alloy is formed. It ranges
from about '00000025 down to zero and may even be negative.
It appears to be impossible to arrange the conditions before-
hand so that the product shall have a particular coefficient
of expansion ; but all pieces from the same ingot have the
same coefficient and it is only necessary to test one specimen
from each ingot when made.

The 45 per cent, alloy has a coefficient equal to that of

E

50 HEAT

platinum, and consequently can be sealed through glass. In
view of the increasing difficulty of obtaining platinum this
is most important.

Many of the compensating devices already considered have
been rendered obsolete by the discovery of invar. An invar
tape stretched under a constant known tension is much used
in survey work ; the tape hangs in a curve of definite shape
and the distance between two marks near the points of
support is determined beforehand on a comparator.

Another important use is in clock pendulums. Even in
the case of the variety of invar with the largest coefficient of
expansion, the ratio of the coefficient of invar to that of iron
is about the same as that of iron to mercury, and consequently
a pendulum can be made on Graham's principle with an invar
rod and an iron bob.

The bob is made in the form of a cylinder, bored so that it
slips over the invar rod, and rests on a nut screwed on the
lower end of the rod. By making the cylinder of the right
length, exact compensation can be obtained, and this con-
struction is now very largely used. It is more satisfactory
than using a variety of invar with a zero coefficient for
pendulum construction, as the pendulum can be over compen-
sated for expansion so as to balance the other slight errors
of the clock, to which reference has already been made.

M. Guillaume points out that other alloys of nickel steel
may be used with advantage for gauges and scales in engineer-
ing workshops, since by altering the percentage of nickel an
alloy can be made having a coefficient of expansion equal to
that of any varieties of steel with which the workshop may
deal.

CHAPTER V
EXPANSION OF LIQUIDS

In the last two chapters we have been considering the
linear expansion of solids ; that is to say, we were concerned
only with changes of length. When a solid is heated, however,
changes take place in breadth and thickness as well as in
length. Thus if we have a sheet of metal and heat it, the
length and breadth increase and consequently the area of the
sheet increases. This increase is spoken of as " superficial
expansion," and the coefficient of superficial expansion is
defined as "the fraction of its original area by which the
area of the sheet increases for each degree rise in temperature,"
or in other words, the coefficient is equal to.

increase in area per degree J , . , ,

— — — i-^ — , the original area being regarded

original area

as the area at 0° C.

Since the length, breadth and thickness of a solid change
on heating, the volume must change. This is referred to as
cubical or volume expansion, and the coefficient of cubical
expansion is defined as in the other cases as the fraction of its
original volume by which a substance expands for each degree
rise in temperature.

Thus in general we may say that coefficient of (linear,
superficial, cubical) expansion

_ increase in (length, area, volume) per degree
original (length, area, volume)

The Relation Between the Three Coefficients of Expan-
sion of a Solid. — Consider a solid sheet of length a and
breadth b (measured at 0°C.), Let the coefficient of linear
expansion of the material be /. Let the temperature of the

51

52 HEAT

sheet be raised from 0° C. to 1° C. Then increase in length
of the sheet is / x original length = fa.

Hence the new length of the sheet is a + fa = a (1 -{-/).
Similarly the breadth increases by fb and the new breadth
is b + fb = b (!+/).

Hence the new area = a (1 -f /) X b (1 -f /)

= ob (1 + /)»

Now the original area was ab.
Hence increase in area = ab (1 + /)2 — ab.
Now coefficient of superficial expansion

_ increase in area per degree

original area
= ab (1 + /)2 - ab
ab

= (1+/)2 -1
= 2/ + /•.

Now /, the coefficient of linear expansion, is always a small
quantity ; consequently /2 is very small indeed, and for all
ordinary purposes can be neglected in comparison with /.
An example will make this clear. Suppose / were equal to
•00002, which, as we have seen, is a fairly high value. Then
/2 would be -000,000,0004. From what has been said in
Chapter III it will be readily understood that we are not likely
to have a value for / so accurate that we need trouble about
corrections in the sixth significant figure. Hence we may say
that the coefficient of superficial expansion is twice the co-
efficient of linear expansion.

Again, suppose we have a rectangular block of length a,
breadth 6, depth c, measured at 0° C. The volume of the
block is then a x b x c. Now let the temperature be raised
through 1°, the coefficient of linear expansion of the material
being / as before.

Then, as we have just seen, the length becomes a (1 + /),
the breadth 6 (!+/), and similarly the depth becomes
c (!+/).

Thus the volume of the block becomes

a (1 + /) x 6 (1 + /) x c (1 + /} 1 = o&c (1 4- n

EXPANSION OF LIQUIDS 63

Thus the increase in volume = abc (1 -f /)3 — o^.
And coefficient of cubical expansion

increase in volume per degree

original volume
_ abc (I + /)3 - abc

abc

= (1+ /)3 - 1
= 3/ + 3/* + f*.

Now we have seen that /2 can be neglected in comparison
with / ; much more then can /3 be. We can therefore neglect
3/2 + /3 and say that the coefficient of cubical expansion is
three times the coefficient of linear expansion.

EXPANSION OF LIQUIDS.

A liquid has no definite shape of its own, but takes up that
of the vessel containing it ; we can therefore only be con-
cerned with volume changes.

If we place some liquid in a vessel, as in Chapter I (Fig. 2),
and heat it, the vessel expands as well as the liquid, and what
we actually observe is the excess of the expansion of the
liquid over that of the envelope. If we measure this expan-
sion and neglect the expansion of the vessel we are said to be
determining the apparent expansion of the liquid, and we
speak of the coefficient of apparent expansion. To obtain
the coefficient of absolute or real expansion we should have
to know the expansion of the vessel, or else we should adopt
the method explained later on page 62.

In speaking of the coefficient of apparent expansion, it is
usual to assume that the containing vessel is made of glass
unless the contrary is stated.

Methods of Determining the Coefficient of Apparent
Expansion of a Liquid.

I. The Volume Dilatometer. — This instrument consists of
a graduated tube of small bore, open at one end and terminat-
ing at the other in a bulb, so that it is like a large thermometer.

64 HEAT

The liquid is placed in the apparatus so as to fill the bulb and
the lower part of the stem,1 and the whole is placed in melting
ice so that the surface of the liquid is just visible above the
ice. The position of the surface of the liquid is noted as soon
as it remains steady. The apparatus is then placed in a
water bath maintained at some definite temperature, e.g.
50° C., care being taken that the liquid is only just visible
above the water surface. The position of the liquid in the
stem is again noted.

If the area of cross section of the tube is known, the increase
in volume is at once obtained by multiplying the distance
between the two positions of the surface by this area. Divid-
ing this increase by the temperature range (in our case 50°)
gives the mean increase in volume per degree. To obtain
the coefficient of apparent expansion we must divide by the
original volume of the liquid. In order to use the instrument
we must therefore know the volume of the bulb and the area
of cross section of the tube, or what comes to the same thing,
the volume of the tube between each graduation.

The most accurate way of finding these volumes is to find
the weights of some liquid of known density (preferably
mercury, which does not wet the surface), which fill the bulb
and a known length of the stem respectively. This can be
done once for all and the instrument thereby calibrated ; it
is, however, more usual to adopt some form of weight dilato-
meter in which the liquid itself is used in the manner described
below.

II. Laboratory Method using a Specific Gravity Flask. —
A specific gravity flask is weighed ; it is then filled with the
liquid at the temperature of the room (which should be noted) .
The stopper is replaced and the excess of liquid which has
overflowed from the small hole carefully wiped off from the
outside. The flask is weighed again. It is then placed in
a beaker of water maintained at some temperature, e.g.
60° C. It should be immersed to such a depth that the neck

1 The filling is accomplished in the same way as in the case of a
thermometer.

EXPANSION OF LIQUIDS

55

is just above the surface of the water, but the lower end of
the stopper is below (Fig. 25). If the liquid is lighter than

water the flask may have to be
fastened in some way.

The expansion of the liquid
causes some of it to come out
through the opening and this
should be wiped off. When no
more comes out the flask is re-
moved and allowed to cool ; the
outside is dried and the flask
and its contents weighed.

It will be convenient to take
some actual results in order to
illustrate the way in which the
coefficient is worked out.

FIG. 25.

FIG. 26.

Weight of flask empty .

Weight of flask full of turpentine at 15° C.

Weight of flask and turps after heating to 60° C.

15-45 gms
68-30 gms
66-11 gms

Consider the liquid left in the flask. Before heating, this
liquid was not able to fill the flask, but would have filled up
to the level A (Fig. 26). The remaining space was filled up
by 68-30 — 66-11 gms. = 2-19 gms., the weight of the liquid
which has come out.

At 60° C., however, the liquid left in is sufficient to fill the

56 HEAT

flask. (Since we are dealing with apparent expansion we
regard the volume of the flask as constant ; actually, of course,
it is larger at 60° C. than at 15° C.)

Let a represent the volume of the bottle up to A and 6
represent the volume of the rest of the bottle. Then a c.c.
of liquid at 15° C. expand b c.c. when heated from 15° to
60° C. Notice that it is the volume of liquid left in the -flask
which expands 6 c.c. The whole flask full of liquid expands
by the amount represented by the volume of expelled liquid
measured at 60° C., which is bigger than 6, the volume of the
expelled liquid at 15° C.

To determine a and b we have that a c.c. of turpentine at
15° C. weigh 66-11 — 1545 = 50-66 gms., and 6 c.c. of tur-
pentine at 15° C. weigh 2-19 gms. But since we are dealing
with the same liquid at the same temperature the volumes
are proportional to the weights, i.e. a : b as 50-66 : 2-19.

Thus we may say that 50-66 volumes of turpentine at
15° C. expand 2-19 volumes for a rise in the temperature of
45° (a " volume " represents the volume of 1 gm. of turpentine
at room temperature). Thus the expansion per degree is

2-19 ,

volume.

45

To get the coefficient we must divide this by the original
volume at 0° C. Now the original volume at 15° C. was
50-66 " volumes," and since the expansion per degree is

2-19 2-19

— - — volumes, the expansion for 15° C. is — — x 15 = -73
45 45

volumes. Thus the volume at 0° C. would have been 50-66 —
•73 = 49-93 volumes. Thus coefficient of apparent expansior

9.1Q

= — 0 = -00097.

45 x 49-93

Notice that in dealing with liquids such as turpentine, with
a fairly high coefficient of expansion, we can no longer consider
the original volume at 15° C. as being a sufficiently near
approximation ; we must either start the experiment at
0° C. or else deduce the volume at 0° C. from our readings.
The latter course is preferable in the experiment described,

EXPANSION OF LIQUIDS

67

for if we filled the flask at 0° C. and then attempted to weigh
it at the room temperature the liquid would be oozing out
while the weighing went on.

The principle of the above method is sound, but a specific
gravity flask is not suitable for accurate work as the overflow
of the liquid is uncertain ; some other form of apparatus must
be used. A form which works well with mercury is the weight
thermometer (Fig. 27). It is filled
with mercury in the manner de-
scribed in Chapter II, i.e. some air
is expelled by heating and the open
end of the tube clipped under clean
mercury in a dish ; some mercury
enters as the bulb cools. The mer-
cury in the bulb is then boiled for
some time to remove gases condensed
on the inner surface of the glass, and
then allowed to cool with the end of
the tube dipping under mercury.

The experiment is performed in
much the same way as in the case of
the specific gravity flask. The bulb
is first surrounded with melting ice,
the end of the tube still dipping
under mercury. The apparatus is
thus full of mercury at 0° C. It is Fi0- 27.

then taken out and weighed, any

mercury which escapes owing to expansion as its temperature
rises to that of the room being caught in a small crucible and
weighed with the rest. The apparatus is then placed in a
bath at any convenient temperature and the expelled mercury
caught in a small dish and weighed. The loss in weight of the
apparatus may also be determined as a check. The calculation
of the coefficient is performed as before, except that the
" volume " at 0° C. is determined directly instead of having
to be calculated.

The weight thermometer is not suitable for general use, as

58

HEAT

J!

it is tedious to fill and almost impossible to clean out. A more
useful form is shown in Fig. 28 ; the tubes are of fine bore
and a mark is made at A. The apparatus is filled and by
judicious tilting the liquid is made to flow out of B until the
surface in the other tube is at A. This procedure is repeated
at the high temperature before taking out the apparatus and

weighing. In any accu-
rate work, water baths
maintained at a constant
temperature by a ther-
mostat must be employed
and the instrument kept
in the bath for a long
time. Such instruments
are known as pykno-
meters.

Although the weight
thermometer is not suit-
able for general use in
determining coefficients it
is useful as means of
measuring temperatures
(hence its name). It is
filled with mercury at 0° 0. and then placed in the liquid of
which the temperature is to be found, and the weight of
mercury expelled is determined. The weight expelled when
the instrument is heated to 100° C. is also found and from
these weights the temperature is found. The method, al-
though troublesome, has advantages in that no calibration of
the tube is required, and uncertainties as to the zero are
avoided as the first weighing is at 0° C.

Absolute or Real Expansion. — The pyknometer gives
us a means of determining apparent expansion. If we knew
the expansion of the vessel we could then apply a correction
and deduce the absolute expansion of the liquid. Now the
absolute expansion of mercury has been determined by the
method, explained at the end of the chapter, and is known

FIG. 28.

EXPANSION OF LIQUIDS 69

with great accuracy. We can therefore standardize any
pyknometer once for all by determining the coefficient of
apparent expansion of mercury in it ; the difference between
this and the coefficient of absolute expansion gives the
correction due to the expansion of the vessel. Mercury is
very suitable for this purpose, as in addition to its other
advantages its coefficient of expansion is small as compared
to such liquids as alcohol or turpentine ; the expansion of the
vessel therefore makes a considerable difference between the
real and apparent expansion of mercury. It may be noted that
for volatile liquids of low density some form of volume dilato-
meter standardized by mercury is better than a pyknometer.

Use of Fused Silica Vessels. — We have already seen
that fused silica has a very small coefficient of expansion,
about the same order as that of invar. The volume expansion
of silica is only about 1 per cent, of that of mercury and much
less than 1 per cent, of that of many liquids. Consequently
for all work except that of the very highest accuracy, a deter-
mination of the coefficient of expansion of a liquid in a silica
pyknometer or volume dilatometer may be regarded as giving
the coefficient of absolute expansion.

The Expansion of Water. — In the case of most liquids
the coefficient of expansion over a moderate range of tempera-
ture is approximately constant : that is to say the increase
of volume from say 10° C. to 20° C. (as measured by a mercury
thermometer) is about the same as that from 20° C. to 30° C.
or 30° C. to 40° C. In the case of water, however, this is not
even approximately true, the coefficient at high temperatures
being much greater than at low. Moreover, if some water
at 0° C. is gradually heated it contracts until its temperature
is about 4° C. and then expands as the temperature rises, so
that its volume at 8° C. is very nearly the same as at 0° C. :
the expansion then continues at an increasing rate as the
temperature rises.

This effect may be investigated by using a volume dilato-
meter with a very fine stem ; but as the volume changes in
the neighbourhood of 4° C. are rery small, it will be necessary

60

HEAT

to use a fused silica vessel to show them directly. With a
glass vessel standardized by mercury the changes can be
deduced from the observations. The following experiment
is an example of a simple and fairly quick way of obtaining
some idea of the magnitude of the effect.

A fused silica pyknometer was filled with water. It was
slightly tilted, so that the water in the tube B (which is of

smaller bore than the
other) ran to the open
end (Fig. 29). The
effect of surface tension
is to prevent the water
coming out unless there
is sufficient pressure,
and the tilt was not
enough to do this but
only to ensure the sur-
face being at the end of
B. A small mark was
made near the surface
of the water in A. The
pyknometer was placed
in a large beaker of
water (about 2 litres)
provided with a very
efficient mechanical
stirrer. This latter is
essential. The pykno-
meter was wired to an upright glass rod held in a clamp ; a
good thermometer reading to 11(y degree being fastened with
its bulb near to it. Fastening was needed to avoid the in-
struments being washed about by the circulating water.

The distance of the surface in A from the mark was read
with a glass mm. scale. Small pieces of ice were added
until the thermometer read about 8° C. ; the temperature
was maintained at this for about five minutes and the distance
of the surface A from the mark measured. More ice waa

PIQ. 29.

EXPANSION OF LIQUIDS 61

added until the temperature fell to 6° C. and the process
repeated.

Readings were taken at 4° C., 2° C., and as near 0° C. as
it was possible to get in a reasonable time. Some of the cold
water was then syphoned out and enough tap water added to
bring the temperature up to 2° C. again. In this manner
readings at 4°, 6°, 8°, and 10° C. were obtained. The whole
experiment took about an hour. The readings were taken
to 0-5 mm., and the figures obtained were as follows : —

Tern perature.

Distance of Surface
from Mark.

Temperature.

Distance of Surface
from Mark.

10-7° C.

4- 1*5 mm.

2*

-.2-0 mm.

8°

— 1-0 mm.

4°

— 25 mm.

6°

— 2-0 mm.

6°

- 2-0 mm.

4°

— 2-5 mm.

8°

— 1-0 mm.

0-7"

— 1-5 mm.

10-9°

+ I'O rnm.

Fig. 30 shows the results graphically.

The diameter of the tube A was found by a micrometer
microscope to be about -9 mm., each mm. length therefore
represents a volume of -1 x n x (-045)2 = -00064 c.c. about.
It so happened that the volume of the dilatometer was about
6-4 c.c., so that each mm. on the tube represented about
•0001 of the volume of water.

It will be seen that the graph is nearly symmetrical about
4° C. ; 1 c.c. of water at 4° C. expands about -00005 c.c. if
heated to 6° C. or cooled to 2° C., and about -00015 c.c. if
heated to 8° C. or cooled to 0° C.1

It may be noted that since the coefficient of cubical expan-
sion of glass is about -000025 the apparent expansion of
water in a glass vessel would not begin until somewhere
about 6°C.

At about 4° C. a given mass of water occupies the smallest

1 More accurate work gives values about -000038 and -00012 respec-
tively.

62

+2

HEAT

0123456789 10 11 t2°C
Temperature.

FIG. 30.

possible volume ; this temperature is therefore taken in
defining the gramme. Much work has been done in finding
the exact temperature of maximum density ; the most
accurate is probably that of Joule and Playfair.1 The mean
of their experiments gave the temperature as 3-95° C.

The Determination of the Absolute Expansion of Mercury.—
The first experiments on the expansion of mercury in which the
expansion of the vessel is not involved * were performed by
Dulong and Petit about 1816, and their method was improved
by Regnault in 1842. Their method consisted in comparing the
densities of mercury at different temperatures. If the density
of a liquid at two different temperatures is known the expansion
between those temperatures can be at once worked out. For
suppose we knew that the density of a certain liquid at 0° C. was
•667 gms. per c.c. and at 100° C. was «65 gms. per c.c., we could
proceed as follows : —

1 Joule's Scientific Papers. Vol. II.

' bee note at the end of the chapter on the expansion of glass vesssel.

EXPANSION OF LIQUIDS 63

If we weigh out say 50 gms. of the liquid, we shall have 50 gms.
of it whatever the temperature ; the mass will not alter. Th*
volume will however vary. Thus at 0° C. the liquid would

50 60

occupy jgg=- = 75 c.c. : at 100° C. it would occupy ^= = 77 c.c.

Thus our liquid would occupy 75 c.c. at 0° C. and 77 c.c. at 100° C. ;
we know then that 75 c.c. at 0°C. becomes 77 c.c. at 100° C., and
with a different mass of liquid the volumes would be proportional.
We do not even require to know the densities separately, but only
their ratio, since for any given mass X

volume at 0° C. density at 0° C. density at 100° C-
volume at 100° C. = X ""' density at 0° C.

density at 100° C.

The problem, therefore, resolves itself into finding some means
of comparing the density of a liquid at different temperatures.

Dulong and Petit adopted the U tube method. The principle
of this method (which may be found in any elementary textbook
of hydrostatics) may be briefly illustrated as follows.

Let Fig. 31 represent two vertical tubes of different diameter
connected by a horizontal tube. If any liquid is put in it will
stand at the same level in each tube (neglecting the effect of
surface tension, see page 66). The two upright columns of
liquid therefore balance. They are not of the same weight (for
the tubes are of different size) but the hydrostatic pressure, i.e.
pressure per unit area at the base of each column, is the same,
otherwise the force across a section A from left to right would not
be equal to the force from right to left and movement would take
place.

If the liquid in the two upright tubes were of different densities,
the two columns would balance when they produced equal hydro-
static pressure at their bases. Let the height of the right-hand
column be h and that of the left-hand column h'. The pressure
per unit area due to a column of height h is simply the weight of a
column of liquid of unit cross section and height h, and does not
depend on the size or shape of the tube (cp. the well-known
" hydrostatic paradox " of Pascal). Hence if the two columns
balance we know that although their weights may be different
the weight of a column of the right hand liquid of unit cross-

64

HEAT

section and height h is equal to the weight of a column of the left

hand liquid of unit cross-section and height h'.

These weights are obviously h x weight of unit volume of

right-hand liquid and h' x weight of unit volume of left-hand

liquid.

h weight of unit volume of left hand liquid /»'

h' ~~ weight of unit volume of right hand liquid ~~ p

where p and />' are the densities, since the weights of unit volumes

FIG. 31.

FIG. 32.

are proportional to the densities. Thus whatever the size and
shape of the tubes, we know that when the two columns balance

h p'

f~, = — . Dulong and Petit used a U tube of the shape shown in

Fig. 32 ; the upright tubes were 55 cm. long, and the horizontal
tube was of narrow bore. One vertical tube was surrounded by
an iron jacket containing pieces of ice and the other by a copper
jacket containing oil which could be heated by a furnace built
round the jacket. The amount of mercury was such that it could
just be seen above the cover of the oil jacket, while the surface in
the cold limb could be viewed through a small opening in the iron
jacket.

The temperature of the oil was given by an air thermometer
and also by a mercury weight thermometer ; * the bulbs of these

1 It might appear absurd to measure the expansion of mercury over
a rise of temperature which was itself determined by a mercury ther-

EXPANSION OF LIQUIDS 65

were made almost as long as the limb of the U tube, so as to give
as nearly as possible the mean temperature of the oil.

As soon as a steady state was reached, the difference between
the levels of the mercury surfaces was determined by a catheto-
meter reading to *g mm. The height of the cold column was also
determined. Experiments were performed with the oil at various
temperatures, the highest being 350° C. At the higher tem-
peratures the indication of the mercury weight thermometer
differed from that of the air instrument ; the latter was taken to
be correct in these cases. In this way the density of mercury at
various temperatures was compared to its density at 0° C.

The coefficient of expansion was then deduced as already
explained on page 63. An example will make this clear. Sup-
pose that with the oil at 100* C. the difference in the heights of
the columns was -925 cm. and the height of the cold column was
50-65 cm. Since the mercury had settled to a steady state before
this measurement was made it follows that the hot and cold
columns balance. So that whatever the size of the tubes
density of the hot mercury 50-65
density of the cold mercury "" 51-576

This is all the experiment tells us.

But consider any given mass X gms. of mercury.

If the X gms. occupy V0 c.c. at 0° C. and V100 c.c. at 100° C.
we have already seen that

V0 Pio o _ 50-65
V10o= /»o "51-575

Since we are dealing with a constant mass X gms. we can say
V0 c.c. at 0° become V100 c.c. at 100°, so that the coefficient

expansion per degree V10o — V«
of expansion is- original volume = yo x 100 "

_L(Vioo-v«A _ _L fZioo \ _L (5!^Z5 _ A

100 \ V0 / ~ 100 \ V0 */ 100\50-65 V

_1 /51-575 - 50-65X 1 -925 •
8 fOOV 50^5 } = 100 * 6fr65 '

This is the coefficient of real expansion, since the measurements
were performed after the tube had done expanding and under

mometer. Dulong and Petit expressly state that the object of this
part was to detect irregularities produced by the expansion of th«
glass of the weight thermometer.

W

66 HEAT

conditions where the size of the tube does not affect the ratio of
the heights.

Regnault's Improvement of Dulong and Petit's Method. —

Regnault (1842) designed an apparatus to avoid certain defects
of the older method. These defects are : (1) the surface of the
hot column of mercury had to project above the cover of the oil
bath in order to be visible, and consequently the upper part of
the column was cooler than the rest ; (2) the arrangement
involved an error due to surface tension. The effect of surface
tension is to make a liquid behave as if its surface formed an
elastic membrane (although no such membrane exists).

The surface of mercury in a tube being convex, the effect of
surface tension will be to increase the pressure at the bottom of
the column and this effect will be considerable if the tube is
narrow.

Now surface tension diminishes very rapidly as a liquid is
heated, and consequently the effect due to the surface of the hot
column of mercury will not balance that due to the surface of the
cold, and an error will be introduced. This is diminished by
making the tubes as wide as possible, which Dulong and Petit
did. If, however, the tube is very wide, the glass must be fairly
thick, which does not conduce to accuracy in reading, and the
weight of the mercury becomes unmanageable unless the length
of the limbs is small. Regnault therefore designed an apparatus
in which the two mercury surfaces observed were at the same
temperature.

His apparatus is shown diagrammatically in Fig. 33.

The upright limbs AB and CD were of iron, about 150 cm.
long and 1 cm. internal diameter. The connecting tube AC was
about 3 mm. diameter, as were BF and DE. The tubes GF and
LE were of glass and were connected together, and to a pipe
which led to a reservoir of compressed air. The cross tube AC
had a small hole at K so that the mercury in AC was exposed to
atmospheric pressure. The limb AB was surrounded by an oil
bath provided with a stirring apparatus and a constant volume
air thermometer P, the bulb of which extended the whole length
of AB. CD was surrounded by a jacket through which cold
water was run and cold water was made to trickle over the hori-
zontal tubes. The pressure of the air in the reservoir was adjusted
90 that the mercury in GF was nearly at the bottom of the tube,

EXPANSION OF LIQUIDS

67

there being sufficient mercury in the whole apparatus to ensure
that the tube AC was full up to the hole K. The tube AC was
made accurately horizontal and the heights H, H', h and h'
measured by cathetometers.

Let T be the temperature of the cold mercury and T" that of
the hot.. Let us further suppose that the temperature of the

•- H

Fio. 33.

mercury in the two limbs G and L is also T ; if it is not a cor-
rection will have to be applied, but it will be small since the
temperature (which is indicated by the thermometer L) will not
differ much from T.

Since the points A and C are at the ends of a horizontal tube

68 HEAT

the pressures there will be the same. Let P be this pressure ;
owing to the hole K this will be practically that of the atmo-
sphere. Let p be the pressure of the air from the reservoir.
Consider the limb LEDC. We have p + column of mercury NE
balancing P + column DC.

Now the hydrostatic pressure due to a column of liquid of
height h is h X weight of 1 c.c. of liquid. If d is the density of
mercury at T, the pressure due to NE is hdg dynes per sq. cm.,1
pressure due to CD is Hdg dynes per sq. cm.
. • . P -f Hdg = p + hdg or

p-P =Hdg -hdg (1)

Now in the limb ABFG, P + column AB balances p + columnMF.
If d' is the density of mercury at T' we have

P + H'd'g = p + h'dg (since column MF is at T°)

or p - P = H'd'g - h'dg ' (2)

Thus from (1) and (2) we have

Hdg -hdg = H'd'g - h'dg
or Hd - hd = H'd' - h'd
or Hd - hd + h'd = H'd'

- 1 H/

d' H - h + h'

Knowing the ratio of the two densities we can get the coefficient
of expansion as in the case of Dulong and Petit's experiment.

Note on the Expansion of Glass Vessels — An approximate
correction for the expansion of a glass vessel can be deduced from
the coefficient of linear expansion of glass. The increase in
capacity would be equal to the expansion of a lump of glass having
a volume equal to that of the interior of the vessel. For imagine
the vessel to be a solid piece of glass ; when heated uniformly all
parts of the glass expand freely and consequently if we consider
an outer layer of the thickness of the walls of the vessel, it must
expand in such a way that there is room for the rest of the glass
inside to expand its proper amount. Thus the increase in capacity
of the vessel per degree should be equal to the capacity at 0° C.
X coefficient of cubical expansion of the glass. But coefficient
of cubical expansion = 3 x coefficient of linear expansion.

Unfortunately this correction is not accurate. For in blowing
glass vessels the physical state and composition of the glass are

1 Since d is the mass of 1 c.c. the gravitational force is dg dynes or
d gros. weight at the particular place.

EXPANSION OF LIQUIDS 69

altered and are neither uniform nor similar to what they were
before heating. Consequently the expansion of the glass before
being worked does not accurately represent the behaviour of the
finished vessel. An attempt to get over this has been made, in
which the bulb was made from a long piece of glass tube of which
the coefficient of linear expansion was determined and the end
then sealed. Although this is fairly accurate the better method
for all ordinary work is to standardize a vessel by using Regnault's
values for mercury.

Silica Vessels. — The cubical expansion of silica is only about
1 per cent, that of mercury ; a determination, of the expansion
of mercury in a silica vessel can be corrected for the expansion
of the vessel, and if the latter had an error of as much as 10 per
cent, it would only mean an error of 1 in 1000 in the value of
the coefficient for mercury. It is not likely that the error would
be as much as 10 per cent, if the expansion of the vessel were
deduced from a measurement of the linear expansion of a sample
of silica by the same maker.

CHAPTER VI
EXPANSION OF GASES

The measurement of the expansion of gases differs from
that of the expansion of liquids and solids in one important
particular. We can say that a given liquid or solid has a
certain volume at any one temperature, but in the case of a
gas such a statement would have no meaning. For a gas
will fill any vessel into which it is put ; if the volume of the
vessel is increased the gas continues to fill it, but does not
exert so great a pressure on the walls. In the same way,
if the volume is diminished, the gas is compressed and exerts
a greater pressure. Whenever, then, the volume of any
quantity of gas is to be measured, we must take into account
the pressure in order that our measurement may have any
meaning. It will be convenient to consider the connection
between pressure and volume before dealing with the changes
due to rise in temperature.

The subject was investigated by Boyle in 1662. The
principle of his experiments was to take a quantity of gas
and compress it, and note the pressure when the volume was
halved. He found that at this stage the pressure was double
that at which the experiment started. He then continued
until this new volume was again halved, and found that at
this stage the pressure was again doubled and so on. These
results are summed up in the statement known as Boyle's
Law (sometimes known as Marriotte's Law) : "At constant
temperature, the pressure exerted by any given mass of gas
is inversely proportioned to the volume."

This Law may be conveniently studied by means of the
apparatus shown in Fig. 34. A stout glass tube AB is fitted

70

EXPANSION OF GASES

71

at the upper end into a pressure gauge, and at the lower
end into an iron tube BC bent into a U. The end C is closed
by a removable screw cap fitted with an ordinary bicycle
tyre valve. The pressure gauge is graduated to read the
" absolute " pressure in pounds per square inch, i.e. when
exposed to ordinary atmospheric pressure it reads just under
15 Ib. per square inch.

The cap C is removed, and water poured into the tube BC ;
the water should practically fill the part BC. In this way
a quantity of air is enclosed in the tube AB. The cap C is
then replaced.

By means of an ordinary bicycle pump the water in CB
can be driven down, compressing the air in AB. The tube
is graduated to show the volume of this air, and the pressure
is given by the gauge. A series of readings are taken and
entered in tabular form thus : —

Volume of Air
cubic inches.

Pressure
pounds per sq. inch.

Pressure x
Volume.

Now if Boyle's Law represents the facts, the pressure should
vary inversely as the volume, i.e. : —

P should vary as ^

or P 4- y should be constant

i.e. P x V should be a constant.

The last column will show the results obtained by multiply-
ing the volume by the corresponding pressure in each
case.

Another form of apparatus for the same purpose is shown
in Fig. 35. It is much more like the apparatus used by

72

HEAT

Boyle, and in the hands of a fairly skilled observer gives
more accurate results than does the method of the preceding
paragraph.

It consists of a glass tube AB closed at the upper end,
and having a length of thick rubber tubing C attached to it.

A.'

• e

s\

U

V

FIG. 34.

FIG.

The other end of this rubber tube is attached to a glass tube
D, open at the top and fastened to a block which can slide
up and down the stand. The tube AB is graduated, and
contains dry air or other gas. C is filled with mercury, which
rises some way into D.

Starting with D near the bottom of the stand, the volume

EXPANSION OF GASES

73

of tne air in AB is noted. The level of the mercury in D
and in AB is noted on a vertical scale ; the difference between
these two gives the difference between the pressure in AB
and that of the atmosphere. This latter is obtained by
reading the barometer. The pressure in AB is thus obtained
by adding to the barometer reading the difference between
the levels of the mercury in the two limbs (or subtract-
ing this difference if the reading in D is below that in
AB).

The block carrying D is then moved up a little way and
readings taken again. In the same manner a series of readings
is obtained and entered in tabular form thus : —

Vol. of gas.

Level of
Mercury
inAB.

Level
inD.

Height of
D above
AB.

Pressure.

Pressure
x VoL

This method has the advantage that the pressure is measured
in terms of a column of liquid (which is more accurate than a
gauge) and the gas is dry ; on the other hand, it is not possible
to reach such high pressures as in Fig. 34 without making the
apparatus an unmanageable size.

It is now known that Boyle's Law is not strictly true for
any gas, especially at high pressures. Gases like hydrogen,
oxygen, air, etc., which are difficult to liquefy, obey the law
very nearly indeed ; in the case of the easily liquefiable gases
the departure from the law becomes considerable (see Chapter
XVII).

The Coefficient of Expansion of a Gas at Constant
Pressure. — To obtain a result comparable with the co-
efficients of expansion of solids and liquids we must ensure
that the volume changes are due to changes of temperature

74

HEAT

only ; in order to do this, the pressure of the gas must be
kept constant throughout the measurements. A convenient
laboratory method of determining the coefficient of expansion
is as follows : —

The closed limb of the
U-tube A (Fig. 36) termin-
ates in a bulb. Some air
is confined in the bulb and
part of the limb by con-
centrated sulphuric acid
which serves the double
purpose of indicating the
pressure of the air and
acting as a drying agent.
The volume of the air is
shown by graduations on
the limb which read directly
to -1 c.c. The U-tube is
surrounded by a wide tube
C which is filled with water,
preferably cooled by ice to
0° C. The open end of A
is above the water surface.
The water is thoroughly
stirred and sulphuric acid
is poured into the open end
of A or run out by the tap
B until the level is the
same in each limb. The
volume of the air id read
off. The water in C is then
heated by blowing in steam

through D for a short time ; it is thoroughly stirred, acid is run
out from B until the level in the two limbs is again the same,
and the new volume of the air noted. A series of readings
are taken at different temperatures, the acid in the two
limbs being brought to the same level for each reading.

Fia 36.

EXPANSION OF GASES 75

The following figures from an experiment illustrate what
happens : —

Temperature. Volume.

10 5° C. ^ 12. 2845 c.c. ^ ,„-

23° C. <^'5 29-7 c.c. > If

51° C. >H 32-5 c.c. > I*

73° C. 34-7 c.c. >

The readings may be plotted in the form of a graph ;
examination of the particular values given shows that
increase in volume per degree ia constant and happens to
come out as -1 c.c. per degree.

To determine the volume at 0° C. we have —
Change for 10-5° = 1-05 c.c.
Volume at 0° = 2845 - 1-05 = 274 c.o.
Coefficient = -1/274 = -00365.

We have neglected the expansion of the glass ; as the
coefficient of cubical expansion of glass is about -000025 it
would only affect the last figure, and so for this sort of ex-
periment may be left out. The apparatus has two great
merits — the use of sulphuric acid (cp. page 83), and the fact
that both surfaces of the acid are in the jacket.

Careful experiments on the coefficient of expansion of gases
were made by Dalton and Gay Lussac, and later Regnault
carried out a series of very accurate determinations. The
results show that to a very close approximation the coefficient

is the same for all gases,1 being equal to -00367 or — — . This

273

fact is known as Charles' Law (sometimes as Gay Lussac's
Law) and may be stated thus : At constant pressure a gas

expands — — of its volume at 0° C. for each 1° C. rise in
273

temperature.

The Pressure Coefficient. — If we heat a gas but do not
allow it to expand, it is clear that the pressure will increase.

1 Regnault gives the value for hydrogen as 07^.1 &Bd for air as
1

76

HEAT

In this case we can measure the coefficient of increase of pres-
sure at constant volume, by which we mean the fraction of its
value at 0° C. by which the pressure increases for each degree
rise in temperature. This coefficient is often referred to as
the pressure coefficient and denoted by ft to distinguish it from
the coefficient of expansion at constant pressure, which is
spoken of as the volume coefficient and denoted by a.

If the gas obeys
Boyle's Law accurately
a and ft are equal. For
let the volume of the
gas at 0° C. and pressure
P0 be V0. If the
temperature be raised
to T° C. and the pres-
sure kept constant the
volume will become V
where

V = V0(l+aT). .(1)
If now the temper-
ature be kept at T° C.
and the gas compressed
until its volume is once
more V0, the new pres-
sure P will be given by
Boyle's Law equation

P0V = PV0.
From (1) we have

P0V0(1 + aT) = PV0
or P0(l + aT) = P.
But if we had through-
out kept the volume at

37

V0 we should have had by definition of ft.
P0(l + /3T) = P.

Thus a = ft.

Determination of ft for Air. — A convenient laboratory
method is by means of the apparatus shown in Fig. 37.

EXPANSION OF GASES

77

The bulb is connected by a glass tube of small bore to the
wider tube C which is connected by thick rubber tubing to B.
B is carried by a block which can move up and down, in the
same manner as the apparatus for verifying Boyle's La\*
described on pages 72 and 73. (That apparatus is in fact
often provided with a separate bulb so that it can be used
for this experiment as well as for Boyle's Law.) The bulb
is filled with dry air, and is surrounded by melting ice. B is
moved until the surface of the mercury in the left-hand limb
is just level with the mark. The difference between the level
of the mercury in B and C is noted on the scale ; the pressure
of the air in the bulb is thus obtained by adding (or sub-
tracting if B is below C) this difference to the reading of the
barometer. The bulb is then placed in a vessel containing
water, which is maintained at the boiling-point. B is raised
until the surface of the mercury in C is once more at the mark,
and the difference in level is again noted. From these two
readings the increase in pressure is obtained and B calculated,
as shown in the following experiment : —

Barometer. 74-2 cms.

Temp.

Reading of
Mercury at the
Mark in C.

Reading of
Mercury in
Open Limb.

Difference
in Height.

Pressure
Cm. of Hg.

o°c.

18-0 cm.

18-3 cm.

+ -3

74-5

98-5° C.

18-0 cm.

44-7 cm.

+ 26-7

100-9

Increase in pressure for 98-5° C. = 26-4 cm. of mercury

per degree = -268 cm.
0 = -268/74-5 = -0036

In this experiment also the expansion of the bulb is
neglected. The air in the tube D is also not at the same
temperature as that in the bulb, but the correction for this
is small as the tube can be of small diameter and therefore

78

HEAT

FIG. 38.

the total volume of the air is small compared to that in the
bulb, and in an ordinary laboratory experiment can be
neglected.1 There is, however, the error
due to the fact that the mercury near
D is at a different temperature from
that in B and consequently has a
different surface tension which produces
a considerable error in a narrow tube
(cp. page 66 on the absolute expansion
of liquids).

For this reason in Regnault's experi-
ments the manometer was arranged as
shown in Fig. 38. The mark against
which the mercury surface was main-
tained being the extremity of D, and
the manometer being kept in a water
bath, and the pressure adjusted by
pouring in or running out mercury.
A correction for the small volume of
air in the manometer and tube was made.

Gas Thermometers. — Gases such as air, hydrogen, or
nitrogen, can be used as thermometric substances either by
keeping the pressure constant and observing the volume change,
or by observing the pressure necessary to keep the volume
constant. The latter method is the more satisfactory, and
the gas thermometers in actual use are almost invariably of
the constant volume type. Fig. 39 shows the standard
constant volume hydrogen thermometer at the Bureau
International des Poids et Mesures.

Gas thermometers have certain great advantages. They
are very sensitive, the change in pressure per degree being
large, and moreover pressure is easily measured. They
have also a great range of t&mperature, since hydrogen,
for example, remains a gas down to very low tempera-
tures, while the upper limit is practically fixed by the

1 In the form supplied by many makers and used above the tube is
very long and an appreciable error results,

EXPANSION OF GASES

79

material of the bulb. By
tures up to 1100° G. can be
however, certain dis-
advantages. In the
first place they are
necessarily rather large
and cumbersome and
can only be used in
one position. In the
second place, gases
being bad conductors,
a long time must
elapse before the gas
takes up the tempera-
ture of the substance
with which the bulb
is in contact. The
occasions on which a
gas thermometer could
be used directly are
in fact very rare ; their
great use is as a stand-
ard. The tempera-
tures are measured in
the first place by a
mercury-in-glass ther-
mometer or one of the
electrical methods (see
Chapter XVI^ and the
mercury thermometer
or other instrument is
calibrated by com-
paring it with the
standard gas thermo-
meter. This will be
referred to again later
(see Chapter XVI).

using porcelain, tempera-
determined. They have,

FIG. 39.

80 HEAT

Absolute Temperature on the Gas Scale. — A gas

at constant pressure expands — - of its volume at 0° C. for

each degree rise in temperature. Suppose we have some gas
at 0° C., and we begin to cool it. If we cool it 10°, it will

contract — of its volume at 0° C. Suppose we were to

cool it through 273° ; apparently it would contract — — of

its volume at 0° C., i.e. its volume would be zero. Now we
cannot imagine that the gas would disappear altogether,
and so it follows that before we get to — 273° C. something
will happen to make the law break down, e.g. the gas will
liquefy. It follows, then, that whatever gas we use will
cease to obey Charles' Law, and presumably become liquid
before — 273° C. is reached. For this reason — 273° C. is
spoken of as " Absolute zero."

We can carry this idea somewhat further with very useful
consequences. Let us imagine, for the sake of clearness,

t \

A B

FIG. 40.

that we have some gas enclosed in a long straight tube of
uniform bore, closed at one end (Fig. 40). Let the other end
of the tube be open to the air, and let the gas in the tube
be separated from the air by a pellet of some liquid.

Let the temperature of the gas be brought to 0° C., and
let A represent the position of the left-hand edge of the pellet
when this is done. If now we mark off the length of the tube
from A to the closed end into 273 equal parts, each of these

divisions will represent — - of the volume of the gas at 0° C.,
273

i.e, the volumo the gas expands for each degree centigrade.
Now mark olf 100 such divisions to the right of A and let

EXPANSION OF GASES 81

the hundredth division be at B. The arrangement now con-
stitutes a kind of constant pressure gas thermometer and each
division corresponds to 1°. If we want it to read centigrade
degrees, we must number A 0 and B 100, because on the
centigrade scale the melting-point of ice is taken as the
starting-point and called zero. But if we had a scale of
temperature in which each degree had the same value as the
centigrade degree but which started at — 273° C., then we
could make our thermometer read temperatures on this scale
by numbering the divisions from the closed end of the tube,
so that A would be 273° and B 373°. This scale is called
the absolute scale and, as will be seen, the volume of the gas
is proportional to its " absolute temperature."

The absolute scale, then, is the centigrade scale starting
from - 273° C. so that 10° C. = 283° A, and Charles' Law
may then be written : " The volume of any gas is propor-
tional to its absolute temperature so long as the pressure
remains constant."

The absolute scale simplifies calculations of the volume of
a gas at different temperatures. For example, suppose we
have collected 500 c.c. of gas at 15° C. and we require to know
what it would occupy at 0° C.

15° C. = 288° A.
0° C. = 273° A.

500 c.c. : volume at 0° C. as 288 : 273,

O7Q
or volume at 0° C. = 500 x . £ c.c.

Zoo

= 474 c.c.

Regnault's Determination of a. — Regnault adopted several
methods, of which the one described in outline below proved on
the whole to be most satisfactory.

A bulb A (Fig. 41) was connected by means of capillary tubing
B to a manometer CE, of which the limb C was graduated. The
manometer contained mercury. The bulb was filled wi*ih car«^
fully dried air (see page 83) and was then surrounded by melting
ice. Mercury was poured into the limb E of the manometer until
the surfaces in C and E were at the same level as shown. The bulb

o

82

HEAT

was then surrounded by steam ; the air expanding pushed the
mercury down in C and up in E. By opening the tap D mercury
was run out until the surfaces were once more level in some posi-
tion lower than C. The volume of the bulb and capillary tube
had been previously determined, and the expansion is given by
the difference in the two readings on C.

Two corrections are necessary. In the first place, the air in
the part C of the manometer (and in the part of B not in contact
with the ice or steam) is not at the same temperature as that in

FIG. 4i,

the bulb. The volume of the air in B is small, but the amount in C
between the two positions is quite large, about one-third of the
volume of the bulb. In the actual experiment the manometer
was kept in a water bath and the temperature noted. The
quantity of air was also adjusted so that the first position of the
mercury surface was in the narrow tube. The other correction
is for the expansion of the bulb and tube.

The great difficulty in all experiments on the expansion of
gases is to get rid of the film of moisture which is present on the
interior walls of the containing vessel, and if this be not done, the

EXPANSION OF GASES 83

moisture becomes partly converted into vapour when heated, and
makes the gas appear to expand more than it really does ; for this
reason the earlier values of the coefficient were too high. In
order to dry the bulb it is alternately exhausted and filled with
dry air, the process being repeated many times, and the bulb
continually gently heated.

CHAPTER
HEAT AND ITS MEASUREMENT

In Chapter I it was stated that we have the idea of some-
thing being communicated to a body when it becomes hotter
or given out by the body when it becomes colder, and that
to this " something " we give the name heat. We must
now follow up this idea, and we shall see that heat is a quantity
capable of definite measurement. For this purpose it will
be convenient to start with a source of heat and consider
some of its effects. A gas-burner is one source, but it will
be much easier to take an electric heater of the immersion
type, such as is sometimes sold for heating shaving water,
etc. We can put this in liquids and see what it is doing ;
the experimental difficulties in the case of a gas-burner are
much greater (see page 89).

To get an idea of the power of the heater, suspend it in a
beaker containing a known mass of water. Switch on the
current and note the temperature of the water every half-
minute ; the water must be well stirred all the time, other-
wise the observed temperature will depend on the position
of the thermometer. Continue until the temperature is about
50° C. The following results of an experiment l will illustrate
the effect : —

1 The heater used was an ordinary domestic immersion heater
by the G.E.C., working on a 220 volt circuit and taking about 1-5
amperes.

84

HEAT AND ITS MEASUREMENT

Mass of water 830 gms.

85

Crude alcohol, mass 830
gms. (volume = 1,000 c.c.)

Time
(minutes).

Temperature.

0

16-5° C.

1

i

17-5° C.

2

1

19-5° C.

2-5

li

22° C.

3

2

25° C.

3

2*

28° C.

3

3

31° C.

2-5

Si

33-5° C.

3

4

36-5° C.

3

4|

39-5° C.

3

5

42-5° C.

3

H

45-5° C.

2-5

6

48° C.

3

6|

51° C.

2-5

7

53-5° C.

Time
(minutes).

Temperature.

0

16-5° C.

1

1

17-5° C.

3

1

20-5° C.

5

4

25-5° C.

3-5

2

29° C.

4

2*

33° C.

4

3

37° C.

4

H

41° C.

3

4

44° C.

4

4|

48° C.

3-5

5

51-5° C.

It will be noticed that after the first minute l the rate of
rise of temperature is practically constant, being just under
6° per minute on the average. If the experiment be repeated
with the same heater and beaker, but using an equal mass
of alcohol instead of water, it will be found that although

1 At the start the current has to raise the temperature of the coils
and container of the heater before it can communicate heat to the water.

86 HEAT

the volume of alcohol is greater, the rate of rise in temperature
is greater. Table 2 gives the results of such an experiment.
In the interval of time from the first minute to the fifth
the temperature of the water is raised from 19-5° C. to 42-5° C.,
i.e. through 23°. In the similar interval the temperature of
the alcohol is raised from 20-5° C. to 51-5° C., i.e. through
31° C. Now there is no reason to suppose that there is any
large difference between what the heater supplies in one
interval of four minutes and the other four minutes. But
it is clear that we cannot give any idea of what the heater
has done by saying that in four minutes the heater raises
the temperature of liquid by so many degrees unless we know
what kind of liquid was used and how much. But if this
information is supplied the method can be adopted, and
quantities of heat are expressed in this way. The liquid
fixed upon as a standard is water, and a unit of heat is defined
as the amount of heat required to raise the temperature of
unit mass of water 1°. Since there are more than one scale
of temperature and more than one unit of mass there are
several units of heat in use. For scientific work the unit of
heat is the calory, which is defined as the amount of heat
required to raise the temperature of 1 gramme of water 1° C.1
A unit sometimes met with, especially in works on chemistry,
is the large or kilogram calory, the amount of heat required
to raise the temperature of 1 kg. of water 1° C. Two other
units are used by engineers in this country, i.e. The British
Thermal Unit (B.Th.U.), the amount of heat required to
raise the temperature of 1 Ib. of water 1° F. ; The Thermal
Unit (T.U.), the amount of heat required to raise the tem-
perature of 1 Ib. of water 1° C.

Thus, in our experiment, the heater in four minutes was

1 Strictly speaking, the amount of heat required to raise the tem-
perature of 1 gm. of water 1° C. varies slightly at different parts of the
scale. In accurate work this is allowed for, and the part of the scale
specified in the definition (see Chapter VIII, page 128). For ordinary
work the difference can be neglected. In the case of some liquids
(notably alcohol) there is an appreciable difference in the amount of
heat required to raise the temperature of 1 gm. 1° C. at different parts
of the scale.

HEAT AND ITS MEASUREMENT 87

able to raise the temperature of 830 gms. of water 23° C.
We can say, therefore, that it supplied to the water 23 X 830 =
19,090 calories in this time, or an average of 4,773 calories
per minute. Experiments of this type are the basis of the
method of determining the heating value of fuels (see page
89).

Thermal Capacity. — The thermal capacity of any object
is the amount of heat required to raise its temperature 1°.
For example, the beaker full of alcohol had its temperature
raised 31° C. by the heater in four minutes. But the experi-
ment with the water showed that in four minutes the heatei
supplies about 19,090 calories. Hence the beaker of alcohol
requires 19.090 calories to raise its temperature 310.1 We
should therefore say that the thermal capacity of the beaker
of alcohol was 616 calories. We sometimes speak of the
thermal capacity of a substance ; in this case it is understood
to mean unit mass of it. Thus we should say that the thermal

capacity of our alcohol (since we had 830 gms. of it) was — -

ooO

or -74 calories per gramme.

Specific Heat. — It is more usual to deal with the Specific
Heat of a substance. This is defined as the ratio of the thermal
capacity of any mass of the substance to the thermal capacity
of an equal mass of water, or in other words, Specific Heat
of a substance is equal to —
Heat required to raise the temperature of any mass of it 1°

Heat required to raise same mass of water 1°
It should be noted that the Specific Heat is a number and
does not depend on the units of heat or of mass employed.
If the unit of heat is the calory, the Specific Heat is numeric-
ally equal to the thermal capacity of 1 gm. of the substance ;
for example, the Specific Heat of the alcohol of our experiment
is given by —

1 Since the thermal capacity of a mass of alcohol varies considerably
with the temperature, the 616 calories represents the average thermal
capacity over a range 20° C.-510 C. The specimen of alcohol used waa
very impure.

88 HEAT

Thermal capacity of the alcohol 616 „.

Thermal capacity of the same mass of water 830
There are, however, some units of heat not defined in terms
of water (e.g. work units as explained in Chapter VIII), and
if these are used the numerical value of the thermal capacity
will not be equal to the Specific Heat. Specific Heat must
always be defined as a ratio, and not as a quantity of heat.
The distinction is rather like that between Specific Gravity
and Density.

Hitherto we have been dealing with liquids, but solids also
vary to an even greater extent in their thermal capacities.

Fia. 42.

The following experiment is a rough illustration of this (Fig.
42).

A number of spheres of the same size, but made of different
metals, such as copper, tin, lead, are provided with hooks
so that they can be hung from the arms of a handle of the
shape shown in the figure. By this means they are suspended
in a vessel of boiling water and left there for some time.
They are then lifted out by the handle and placed on a sheet
of paraffin wax carried on the ring of a retort stand ;
the handle is disengaged by a slight turn and removed. The
spheres give out heat as they cool and melt some of the wax
with which they are in contact : they therefore sink into the
wax. This goes on until the spheres have cooled below the

HEAT AND ITS MEASUREMENT 89

melting-point of the wax. It will be found that the spheres
have got to very different depths ; if the wax is of suitable
thickness l the copper sphere will melt its way right through
and fall down, the tin will be partly through and the lead
will make very little impression. It should be noted that the
spheres are of equal size and not of equal mass ; to compare
specific heats we ought to have taken equal masses, in which
case the lead would have been smaller and the difference
in behaviour between it and copper would have been more
marked.2

Calorimetry. — Operations involving the measurement of
quantities of heat are classified under the general term
calorimetry, and any apparatus for making such measure-
ments is called a calorimeter. In the simplest forms the
calorimeter is a vessel of suitable material containing a known
mass of water, to which the heat is communicated.8 The
quantity of heat is measured by observing the rise in tem-
perature of the water as in the case of the electric heater ;
there are, however, certain corrections to be considered. The
main principles may be illustrated by a brief account of an
operation of great practical importance — the determination
of the calorific value of fuels. The calorific value of a fuel
is the amount of heat given out when a definite weight, 1 Ib.
or 1 kg., is burnt ; in the cases of gaseous fuels the value is
usually given as so many units of heat per cubic foot or per
cubic metre.

Determination of the Calorific Value of a Fuel such
as Coal. — The practical difficulty is to make sure that all
the heat given out by the fuel is accounted for. One method

1 With spheres about 1 inch in diameter the sheet of wax should be
about J inch thick or a little more.

2 In all cases of heat measurement we are concerned with masses,
not volumes. The volume of any given substance depends on its
temperature and, at any rate in the case of gases, on the pressure. The
mass does not depend on these conditions and so gives us a constant
quantity with which to work.

3 Other types of calorimeter are dealt with at the end of Chapter
X, see page 167.

90

HEAT

is to use a bomb calorimeter (Fig. 43). A known weight oi
the coal, which has been powdered and dried, is placed in a
small capsule inside a strong steel shell or bomb ; this bomb
has a gas-tight cover, fitted with a valve. The cover is put

FIG. 43.

on, and oxygen is admitted through the valve until the bomb
is filled with oxygen under a suitable pressure.1 The bomb
with its contents is immersed in a known weight of water
contained in a metal calorimeter. This calorimeter is sur-
rounded by a water-jacket shown by the dotted lines. The
temperature of the water in the calorimeter is determined by
a thermometer which should read to y^0 C. and care is taken

1 The pressure depends on the amount of fuel and the size of the
bomb. A usual size is to have a bomb of about 600 o.c. capacity, in
which case about 1 gm. of coal would be used, and the pressure of the
oxygen would be about 25 atmospheres

HEAT AND ITS MEASUREMENT 91

that the calorimeter, bomb and water are at the same tem-
perature ; this is done by keeping the water well stirred and
reading the thermometer every minute until the temperature
is constant. The fuel is then ignited by passing an electric
current through a fine platinum wire lying in contact with the
coal and connected to terminals on the outside of the bomb.
The wire becomes red hot and the fuel then burns completely
in the excess of oxygen, the products of combustion remaining
in the bomb.1 The water in the calorimeter is thoroughly
stirred and the highest temperature reached is noted.

The heat given out by the burning coal has raised the
temperature of the known mass of water through a certain
range, but in addition to this the bomb and the calorimeter
(and also the stirrer and thermometer) have been raised
through this same range ; the effect of the stirring was to
make sure that they all started at the same lower temperature
and that they all finished at the same higher temperature.
A correction must therefore be applied to allow for the fact
that these things were heated as well as the water ; in order
to find the amount of heat given by the coal we must multiply
the rise in temperature by the mass of water -f- the water
equivalent of the calorimeter, bomb, etc. By the water
equivalent of any object we mean the mass of water which
requires as much heat to raise its temperature 1° as is required
to raise the temperature of the object 1°, or, in other words,
the amount of water which has the same thermal capacity as
the object.

Thus, suppose that the calorimeter contained 2,500 gms.
of water, the initial temperature was 18-92° C., and the final
temperature was 20-80° C. Let us suppose that the water
equivalent of the whole apparatus was known to be 700 gms.
Then heat given out by the coal =

(2,500 + 700) (20-80 - 18-92) = 3,200 X 1-88 calories.

The water equivalent can be determined by a direct ex-

1 The products of combustion would have a corrosive action on
steel, so the inside of the bomb must be protected by a lining of enamel
or else of platinum or gold.

92

HEAT

periment (see page 97) ; it is, however, possible to calculate
the water equivalent of any vessel if its mass and the specific
heat of the material are known. For if m is the mass and x
the Specific Heat, we have —
Thermal capacity of the vessel

/. water equivalent of vessel
or water equivalent of any vessel

= x times thermal capacity

of m gms. of water.
= thermal capacity of xm

gms. of water.
= xm gms.
= mass of vessel X Specific

Heat of the material.
Bomb calorimeters are
expensive, and a much
cheaper instrument which
can be made to give good
results is Darling's calori-
meter, Fig. 44. The coal is
contained in a nickel cruci-
ble D, and the combustion
takes place under a kind of
bell jar A, which is im-
mersed in the water. Oxy-
gen is admitted through the
tube O, and bubbles out
together with the products
of combustion through
openings at the base. The
issuing gases communicate
their heat to the water and
also help to stir it. The
writer gets very satisfactory
results with this calori-
meter, but a little practice
in regulating the oxygen is usually necessary to be sure
of success.
Determination of the Specific Heat of a Solid.

The most straightforward methods are those which may

FIG. 44.

HEAT AND ITS MEASUREMENT 93

be classified under the general term of the " method of mix-
ture." The simplest form of it, which we shall now consider,
is rather rough, but gives quite satisfactory results to an
accuracy of 5 to 10 per cent. It is suitable only for
metals.

A lump of metal (some 500 to 800 gms.), preferably in the
form of a thick disc, is provided with a wire handle. It is
first weighed to the nearest gramme and then placed in a
saucepan of water heated on a ring burner. A copper l
calorimeter is weighed empty, then about two-thirds filled
with water and weighed again. It is then placed on three
corks (to avoid as much as possible conduction of heat to
the bench), and the temperature of the water is noted. When
the lump of metal has been in the boiling water of the sauce-
pan for some time (5 or 10 minutes) it is lifted out by the
wire handle, given a quick shake and transferred as rapidly
as possible to the calorimeter and immersed in the water.
The water is well stirred (by moving the lump of metal) and
the highest temperature noted.

It will be convenient to follow out the method of calcula-
tion by using some figures from an actual experiment.

Weight of metal lump ..... 427 gms.
Weight of calorimeter empty . . . .192 gms.
Weight of calorimeter -f- water .... 710 gms.
. • . Weight of water . . . 518 gms.
First temperature of water .... 15^5° C.
Final temperature of water . . . 19-0° C.

The lump of metal cooled through a certain range of tem-
perature, i.e. from 100° C. down to 19° C., this latter being
the final temperature of both metal and water. In doing
this it gave out heat, and this heat was measured in just
the same way as in the case of the fuel, i.e. by finding its
effect on a known mass of water in a calorimeter.

1 Calorimeters are usually made of copper, but aluminium is also
suitable.

94 HEAT

The heat given out by the metal raised the temperature
of the water and calorimeter from 15-5° C. to 19° C.
. * . Heat gained by calorimeter and water —

== (518 -|- water equiv. of cal.) x (19 — 15-5) calories.

The water equivalent of the calorimeter may be got either
by direct experiment or by assuming the Specific Heat of
copper. This latter is about -1 (strictly -095), so that the
water equivalent of the calorimeter is •! x 192 or about
19 gms.

. * . Heat gained by calorimeter and water
= (518 + 19) X 3-5 = calories
= 537 X 3-5 calories.

Now this heat was supplied by the metal in cooling from
100° C. down to 19° C., i.e. through 81°.

Turning now to the metal ; if S is its Specific Heat, the
amount of heat given out by the 427 gms. of metal cooling
1° C. is S times that given by 427 gms. of water cooling 1° C.
That is to say, 427 x S calories.

. • . Amount of heat given out by the metal in cooling through
81° would be—

427 x 81 X S calories.

But as actually measured by the calorimeter it was found
to be 537 x 3-5 calories.
. • . 427 x 81 X S = 537 = 3-5
« 537 x 3-5

The above method is rather rough ; two weak points are : —

(1) A certain amount of hot water is carried over with
the metal in transferring it from the saucepan to the
calorimeter.

(2) The metal is cooling as it is being transferred, so that
its temperature is not 100° C. as it is placed in the calorimeter.
It is not, however, so rough as it might appear at first sight,
provided that considerable masses of metal are used ; in the
case considered the errors involved by (1) and (2) were small
in comparison with the total quantities of heat involved.
Incidentally ^1} tends to make the final result too high, and

HEAT AND ITS MEASUREMENT 95

(2) tends to make it too low, so that we are concerned with
fche difference of these two effects and not their sum.1 But
the result depended on a temperature difference of 3-5° C.,
and with a thermometer graduated in degrees this was the
most uncertain part ; it would have been a waste of time to
try and eliminate (1) and (2) unless a more accurate
temperature measurement was possible.

In more accurate work the substance must be heated in
some form of heater in which it does not come in contact
with water or steam, and arrangements are made for quick
transference to the calorimeter. A very simple method, suit-
able for a fairly rough laboratory determination of the Specific
Heat of a solid which is in the form of small pieces (e.g. copper
turnings, iron nails, etc.), is to place the substance in a glass
boiling tube which is placed in a saucepan of boiling water
or passed through a large cork resting on the neck of a metal
can so that the tube is in the steam of boiling water. The
mouth of the tube is lightly plugged with cotton wool, and
a thermometer is placed with its bulb well covered by the
fragments of the substance. The tube is heated for a con-
siderable time until the thermometer reading is constant.
The tube is then removed, the plug of cotton wool taken out,
and the contents quickly poured into the calorimeter. Various
forms of steam heaters are sold, in which the substance can
be poured out of its tube without removing the tube. Fig. 45
shows a more elaborate arrangement, due to Regnault.2

The substance is placed in a small wire gauze basket,

1 By choosing a suitable size and shape for the lump of metal it is
possible to ensure that the two errors very nearly neutralize one another
on the average of a number of experiments. The figures quoted were
from experiments with an apparatus of a suitable size, and the writer
has made many experiments on the magnitude of this error. This
must not be regarded as converting the method into an accurate one,
but merely as indicating what are suitable dimensions. The experi-
ments were undertaken to find out why the results obtained with a
rough apparatus were consistently so close to the accepted values.
But the experiments showed that the errors were fairly small, so that
the method is quite good enough for work in which an ordinary ther-
mometer is used, graduated in degrees.

* Ann. de Chimie et de Physique, 1840, 2e Ser., torn. 73.

96

HEAT

suspended by a thread in the centre of a steam heater. This
central space is closed at the top by a cork and at the bottom
by a sliding shutter A. When the substance has been at its
steady temperature for some time (e.g. half an hour), the

FIG. 45.

screen B is raised, the calorimeter is pushed under the heater
and the substance quickly lowered into it ; the calorimeter
is then quickly withdrawn. The double-walled screen C,
through which cold water circulates, protects the calorimeter
from radiation from the heater.

Errors Due to Cooling. — We have assumed that the
whole of the heat given out by the substance is measured by
the rise in temperature of the water multiplied by the mass of
water -{- water equivalent of the calorimeter. But as soon
as the water becomes hotter than its surroundings it begins
to lose heat and consequently the final temperature is lower
than it ought to be. To minimize this effect in our rough
experiment the calorimeter stood on three corks ; it can be
made very much less by suspending the calorimeter in a larger
vessel so that there is an air space all round it, and by having
the outside of the calorimeter and the inside of the outer
vessel highly polished (see Chapter XIV). In this case, if the

HEAT AND ITS MEASUREMENT 97

rise in temperature of the water is only 3° or 4° the cooling
effect is small, but in very accurate work it must be allowed
for. The method of correction is given on page 114. It is
sometimes recommended that the space between the vessels
should be filled with cotton wool or felt, but although this
helps to diminish the cooling it introduces great uncertainties
in the corrections and should not be done.

Direct Determination of the Water Equivalent of a
Calorimeter. — A simple method is to weigh the empty
calorimeter and leave it for some time with a thermometer
in it, to allow it to take up the temperature of the room.
Some water, at a temperature of about 25° C. or 30° C., is
got ready and its temperature carefully taken with an accurate
thermometer. It is then immediately poured into the calori-
meter until the latter is about two-thirds full. As soon as it
is in the water is well stirred and its temperature taken with
the same thermometer which was used before. The calori-
meter and its contents are then weighed, to determine how
much water was put in. The following figures may be used
to illustrate the calculation : —

Weight of calorimeter empty . . . .192 gms.
Weight of calorimeter + water .... 660 gms.
. • . Weight of water . . . 468 gms.
First temperature of calorimeter , * . 16-5°C.
First temperature of water . . . . . 25-9°C.
Final temperature of water .... 25-55° C.

Heat lost by water = -35 X 468 = 164 calories.

This heat raised the temperature of the calorimeter from
16-5° C. to 25-55° C., i.e. through 9-05°.
Therefore to raise the temperature of calorimeter 1° requires

164/9-05 = 18 calories.
.*. calorimeter is equivalent to 18 gms. of water.

It should be noticed that since the measurement is based
on a fall of temperature of about -35° C., the thermometer
must read to at least -05° in order to get an accuracy of
anything like 10 per cent. The thermometer used was

H

98 HEAT

graduated in TV°, and could easily be read to half this.
Moreover if the water were much above 30° C. the fall in
temperature due to ordinary cooling during the time of the
experiment would be nearly as much as that due to the
calorimeter ; it is found by experiment that if it is not above
about 25 to 30° C. the cooling is negligible in the short time
taken by the experiment.

Specific Heat of Liquids.

The method of mixtures can be applied in two ways. The
liquid can be placed in a calorimeter and a solid of known
specific heat can be heated and dropped in as already described.
Another method is to place a known mass of the liquid in a
thin metal container and heat this in a heater ; the container
with the liquid is then dropped into water in a calorimeter
as in the case of a solid. In this case the thermal capacity
of the container must be determined separately.

Specific Heat by the Method of Cooling.— This method
gives fairly reliable results in the case of liquids.1 It depends
on the principle that if we have a closed vessel at a higher
temperature than its surroundings it will lose heat by con-
duction, convection and radiation (see Chapter XII) ; the
rate of loss of heat will depend on the area and nature of the
surface and its temperature, and on the external conditions,
but will not depend on what is inside it. The idea is there-
fore to measure the time taken to fall through a given range
of temperature first when the vessel is fuD of water and then
when it is full of the liquid. Since the mean temperature is
the same in each case, the rate of loss of heat (calories per
second) will be the same, but as the thermal capacities of the
contents of the vessel are different the rate of fall of tempera-
ture (degrees per second) will be different.

The experimental arrangements are concerned chiefly in
securing (1) that all parts of the liquid are constantly brought
in contact with the walls of the vessel (otherwise the tem-
perature of the liquid will not be closely related to that of

1 Attempts to use this method for solids, in the form of powder tightly
packed, were not satisfactory. Cp. Regnault.

HEAT AND ITS MEASUREMENT

99

the vessel), this involves adequate stirring ; (2) that the
cooling conditions are the same in each case.

A very convenient laboratory method is to make use of
the Callendar apparatus for determining the mechanical
equivalent of heat (see page 111).

Some water is heated to a temperature of about 30° C.,1
and 250 c.c. are introduced into the brass drum , the special
bent thermometer (which reads in fifths of a degree) is in-
serted and the drum (without its silk belt) is rotated by the
motor at about 100 to 120 r.p.m. The time taken for the
temperature to fall through a certain range is noted, e.g. from
25° C. to 20° C., readings being taken at each degree. The
drum is then stopped, the water emptied out and 250 c.c. of
the liquid (previously heated to about 30° C.) put in. The
drum is then started up and the time taken for the tempera-
ture to fall through the same range noted. If there is time, the
experiment can be repeated with water or alcohol as a check
on the constancy of the cooling conditions. The following
figures from an experiment will illustrate the procedure.
Alcohol. Water.

Temp.

Time.

Time
Interval
for 1°.

h. m. s.

m. «.

25° C.

5 40 45

>1 20

24°

5 42 5

>1 40

23°

5 43 45

>1 45

22°

5 45 30

">2 10

21°

5 47 40

^>2 15

20°

5 49 55

Temp.

Time.

Time
Interval
for 1°.

h. m. s.

m. 8.

25° C.

6 36 0

>2 50

24°

6 38 50

>3 20

23°

6 42 10

">3 50

22°

6 46 0

»s.

>>4 30

21°

6 50 30

""^

^>4 55

20°

6 55 25

1 The thermometer supplied with the apparatus usually reads up to
30° C. ; the water must therefore be heated not much above this
temperature. It will be cooled a little an it ia put in.

100

HEAT

Alcohol.

Temp.

Time.

Time interval for 1°.

h.

m.

8.

m.

S.

25° C.

7

33

35

^> 1

25

24°

7

35

0

^> 1

40

23°

7

36

40

"> 1

45

22°

7

38

25

21°

7

40

30

> 2

5

^> 2

25

20°

7

42

55

Water equivalent of drum = 39 gms. (see page 115).
Density of the alcohol = -83 gms. per c.c.
Taking the mean of the two sets of figures for alcohol, we
have —

Time (seconds).

Temp.

Range.

Alcohol.

Water.

25° 24°

82-5

170

24° 23°

100

200

23° 22°

105

230

22° 21°

127-5

270

21° 20°

140

295

170

2-06

82-5

200

2-00

100
230

2-19

Mean

>. = 2-1

105

270

2-11

nearly.

127-5

295

2-10

140

Taking any of these temperature ranges, say 25° to 24° >
the heat given out when the drum contains water is —

(250 + 39) = 289 calories.
Time taken to do this =170 seconds.

289
Rate of emission of heat = — calories per second.

This is the rate of emission of heat when the mean tern-

HEAT AND ITS MEASUREMENT £ MGfl

j „ ~» ' - -j J " " » "- »*'»••»
perature is 24-5° C., and this rate is 'the -scente whatever the

contents of the drum.

It is therefore the rate of emission when the drum contains
alcohol.

The alcohol took 82-5 seconds to cover the range, and so
the total heat given out was —

- x 82-5 calories.

This is therefore the heat given out by the drum and alcohol
cooling 1°C.

Now the mass of the alcohol is 250 x -83 = 207 gins. So
the heat given out in cooling 1° is —

207 S + 39 calories.
Thus 207 S + 39 = 289 X ~ = 289 X ^-

1 iU .Z'UO

In the same way we could compare the values for any of
the other temperature ranges, and the expression would be
the same except that for 2-06 we should have the ratio of the
times for that particular range. In working out the result
we take the mean value 2-1 for this ratio, and we have —

207 S + 39 = •
Z'L

or 434-7 S + 81-9 = 289

434-7
= 48.

If a Callendar apparatus is not available, an arrangement
such as that shown in Fig. 46 works well.

A is a tin, the lid of which is fitted with a cork through
which pass a thermometer D and a short piece of glas"s tubing
which serves as a bearing for the stirrer B. This is made of
wire gauze soldered to a knitting needle, and driven through
a flexible coupling C (a spiral spring secured with corks) by a
small motor. B rests on a cardboard support F in a larger
tin E, which is surrounded by cold water kept constantly
circulating and which is supported on metal blocks G.

102'

HEAT

E has to be weighted to keep it from floating.
of E is closed by two pieces of cardboard.1

The top

I

D

K

FIG. 46.

Specific Heat of Gases.

The methods of determining these are discussed in Chapter
XIX.
Variation of Specific Heat with Temperature. — It is

found that in general the specific heat of a solid or liquid
is not quite the same at different temperatures. The experi-
ments we have considered have been mostly to determine
the mean value over a range between about 20° C. to 100° C.
In the case of solids the value at high temperatures is in general
greater than at low. Thus for iron the mean specific heat

1 Any form of apparatus for this method must provide for very
adequate stirring and also constancy of the cooling conditions. As
regards the latter, conduction and convection account for the greater
part of the heat loss (at such temperatures as are likely) and the design
of the apparatus must be based on this fact. The rotating Callendar
apparatus is particularly satisfactory from this point of view ; on the
other hand its water equivalent is rather large.

HEAT AND ITS MEASUREMENT 103

over the range 20° to 100° C. is about -12 and the mean
specific heat over the range 0° to 1100° C. is about -15.

It should be noted that the specific heat of a substance
over any range denotes the ratio of the mean thermal capacity
of the substance over that range to the thermal capacity of
water at some particular temperature, usually 15° C. or 20° C.
(see page 128). We do not compare it to the thermal capacity
of water over the whole range, even if we could have water
existing at all the temperatures.

CHAPTER VIII

HEAT ENGINES— THE RELATION OF HEAT AND

WORK

We have seen that one of the effects of heat on most sub-
stances is to cause expansion ; this effect is the basis of what
is perhaps the most important practical application, i.e.
heat engines. Heat engines are devices for obtaining mechani-
cal work by means of heat, and the term includes hot-air
engines, steam engines (including turbines), and internal
combustion engines. A detailed consideration of the develop-
ment of these engines would take up too much space in a book
of this size ; at this stage it will be sufficient to consider certain
broad principles.

Fig. 47 represents an extremely simple form of a hot-air
engine.1 A flask A, containing air, is connected by a short

1 This device is due to Principal E. H. Griffiths and was shown by
him at a course of lectures given at Leeds in 1900, which the present
writer, as a small boy, was privileged to hear ; the lectures are printed
in the form of a book, Thermal Measurement of Energy, published by
the Cambridge University Press.

104 HEAT

tube to a U-tube B containing mercury. The short tube has
a branch closed by a tap C. On placing a small flame under
A the air expands, pushing down the mercury in one limb of
B and raising it in the other. C is then opened for a moment
and then closed ; this lets out some air and allows the mercury
to drop back. The momentum of the mercury causes it to
overshoot its equilibrium position and rise some way up the
nearer limb ; if the amount in the tube is correctly adjusted
the mercury will go on oscillating as long as the flame is
applied to A.1

The operations which go on are as follows : the air takes

1 As the successful working of the engine sometimes presents
considerable difficulty the following details may be useful as time
savers : —

Setting up the apparatus. The quantity of mercury has to be adjusted
to the dimensions of the flask by trial. The flask should not be too
large. The dimensions of the apparatus which I am using at present
are:

Diam. of flask, 7 cms.

Length of U-tube (i.e. vertical height from the bottom of the bend

to the top of the open limb), 21 cms.
External diameter of the U-tube limbs, 2 cms.
Vertical height from bottom of the bend to the top of the surface

of the mercury (when at rest), 14-5 cms.

The U-tube was one which happened to be available and the size of
the flask to work with it was found by trial. The flask is Jena glass
and is heated by a spirit lamp. The flame must play on the glass
surface and with a bunsen burner the engine quickly blows a hole in
the flask on the compression stroke.

Starting. Even when the adjustments are correct the engine will
not always start very easily ; I have found however that the following
method can always be relied upon : — The flask is heated by the spirit
lamp and as it expands the tap is opened and is kept open until
expansion ceases, i.e., until on closing the tap the mercury surfaces
remain at the same level. This is important. The tap is then closed
and the oscillation started by introducing a glass rod (about 1 cm.
thick) into the open end of the U-tube and working it up and down
at about the proper period for the mercury for a few times (one can
easily tell by the feel whether the vibrations are forced or not). On
withdrawing the rod the oscillations should continue ; if they do not
the process can be repeated. The engine will go on indefinitely and
the amplitude can be adjusted by altering the height of the flame.
A range of motion of about 5 cms. is suitable. The period of the engine
quoted above is -8 sees.

HEAT ENGINES 105

in heat and expands, pushing the mercury outwards ; it then
loses heat to the walls of the tube and neck of the flask and
therefore contracts, allowing the mercury to come back.
It then takes up more heat and expands again, and so the
process continues.1
Although this device is merely a pleasing toy and could

FIG. 47.

not be of any practical use, it has the essentials of a heat
engine, i.e. : —

(1) The working substance (air) which expands if heated.

(2) A source of heat (the flame) from which the working
substance takes in heat to enable it to expand.

(3) A cooling arrangement so that when the air has per-
formed its working stroke it can give up heat and so contract
and get back to its original size ready for the next stroke.

It will be noticed that this third stage is essential in order
to get continuous work from the engine. We shall see that
in all heat engines this rejection of heat occurs and it is an
important point ; there will be a good deal to be said about
it later on.

Practical hot-air engines have been constructed using a
cylinder and piston in place of the flask and mercury ; they

1 The mercury has to be adjusted so that its natural period of oscil-
lation in the tube suits the rate of taking in and rejecting heat. The
adjustment is often a matter of some difficulty.

106

HEAT

have an important addition (the displacer), but they work
on this alternate heating and cooling of air. They are not
much used in practice (except for such purposes as running
small stirring apparatus in a laboratory) because they would
have to be very large indeed to develop any considerable power ;
a very large volume of air would be needed to get a reasonable
expansion.

In the steam engine, advantage is taken of the fact that
when water is heated and turned into steam a very large volume

To Boiler

FIG. 48.

change takes place. The volume of steam at 100° C. and
atmospheric pressure is about 1,700 times that of the water
from which it is formed, and this makes it a much more
suitable working substance than air. Fig. 48 represents a
section of an ordinary double-acting slide-valve cylinder.
Steam from the boiler enters the steam-chest C and passes
down the passage Si driving the piston P to the right. The
slide valve V prevents the steam getting into the passage S2.
The slide valve then moves to the position V2 shown by
dotted lines, whereby steam passes from the steam-chest

HEAT ENGINES 107

down S, urging the piston to the left, while the steam left
in the cylinder from the previous stroke passes up S to E
which is connected to an exhaust pipe leading to the open
air (non-condensing engines) or to a condenser. Thus steam
is admitted to alternate ends of the cylinder, and after per-
forming its stroke is let out to the air or to the condenser.

Although the arrangement is different from that of the hot-
air engine, there are the same fundamental operations as before,
i.e. : (1) The working substance (water) takes in heat when
it is in the boiler and is turned into steam ; (2) the steam in
virtue of its pressure, due to the expansion when water
evaporates, performs its working stroke ; (3) the piston is
brought back by allowing the steam to pass out carrying a
large amount of heat with it which is given up to the air
or to the condenser.

The internal combustion engine makes use of the fact that
when a mixture of gas and air or petrol vapour and air is
ignited, a large amount of heat is generated and the gaseous
products of the explosion owing to their high temperatures
occupy a large volume. Fig. 49 represents a simplified section
of the cylinder of an ordinary four-stroke engine with over-
head valves. The mixture of gas or vapour and air comes
into the inlet space I, while E communicates with the exhaust.
There is only one working stroke in every two revolutions,
and the process is as follows : —

As the piston P moves down,1 the valve 1 is made to open
by the valve operating mechanism, and the mixture is drawn
into the cylinder. The valve 1 then closes and the piston
moves up compressing the charge into the top of the cylinder.
As the piston reaches the top the charge is ignited by means
of the sparking plug S, and the force of the explosion drives
the piston down ; this is the working stroke. When the
pisx/on begins to come up again, the valve 2 is opened allowing
the products of combustion to be swept out into the exhaust

E. The operations are then repeated. A heavy flywheel is

•\

1 If the engine is not already running it has to be started by external
agency.

108

HEAT

provided so that its momentum will enable the piston to
move through its three non- working strokes.
As before we have the same three fundamental operations,
(1) The working substance is supplied with heat (from

.e.

the combustion) ; (2) the working stroke is performed ;

Porcelain,
' or mica.

Diagrammatic Section

of
Spark Plug

FIQ. 49.

(3) the working substance is let out, giving up a large amount
of heat to the air.

In these engines the cylinder has to be cooled either by
a water-jacket or by air, and a good deal of heat from the
explosion is ultimately dissipated in this way also.

Work done by an Engine. Brake Horse-power.

Definition of Work. — When a force is applied at a point and
the point of application moves, work is said to be done, and

HEAT ENGINES 109

the amount of work is given by the product of the force and
the distance moved by the point of application, the distance
being measured in the direction of the force.

Thus, if we have a 100-lb. weight and we try to lift it and
fail, no work has been done, although a force may have been
exerted. But if we raise the weight 1 foot, we have done
100 foot-pounds of work ; if 2 feet, we have done 200 foot-
pounds, and so on. If, however, we push the weight hori-
zontally 2 feet, we have not done 200 foot-pounds ; for
example, it may need a force of 20 Ib. weight just to move
the weight against friction, and in this case we shall have
done 2 X 20 = 40 foot-pounds of work. It will be observed
that there is no question of time ; if the 100-lb. weight is
raised 2 feet, we have done 200 foot-pounds however long
we took over it.

Power. — Power is defined as the rate of doing work ; in
engineering it is usually measured in horse-power. An engine
is said to be working at the rate of 1 horse-power if it is doing
33,000 foot-pounds per minute or 550 foot-pounds per second
This number was based on experiments, but, as a matter of
fact, it represents rather an exceptional horse.

In measuring the power of an engine we might set it to
wind up a weight and so calculate the work done in a given
time ; this, however, would be most inconvenient as, unless
the engine is very small, very large weights or very great
heights would be involved. It is more convenient to set the
engine to work against the friction of some sort of brake;
the work done can then be measured as described below.

Fig. 50 represents a rope passing over the rim of a wheel
on the engine shaft. A weight W is suspended at one end of
the rope and a spring balance is attached to the other. If
the wheel is rotating as shown, the friction of the rope on the
wheel will have a tendency to raise the weight ; the difference
between the weight and the reading of the spring balance will
give the frictional force, since the frictional force and the
spring balance between them support the weight.

Thus, suppose that the weight were 20 Ib. and the spring

13 Ib. wt. represents the

110 HEAT

balance read 7 Ib. Then 20 — 7 =
frictional force.

Now consider the portion AB of the rope which is in con-
tact with the wheel. If this isolated portion were pressed
against the wheel so that the frictional force remained the
same, i.e. 13 Ib. wt., and the wheel were at rest, the work
done if the piece were hauled once round the wheel by means
of the end A would be 13 Ib. X distance A moves, i.e. 13 X

Spring
balance

(a)

(b)

FIG. 50.

circumference of the wheel (strictly the circumference of the
circle formed by the central line of the rope).

The same work is done if AB remains at rest and the wheel
turns one revolution ; hence in our actual case the work
done against friction per revolution is 13 x circumference of
wheel (neglecting thickness of rope). So that in order to
find the work done per minute, all we have to do is to read
the spring balance and find by a revolution counter (or a
tachometer) the number of revolutions per minute. Then
work done per minute = (W — spring balance reading) x
circumference of wheel in feet X number of revolutions per
minute. This value divided by 33,000 gives the horse-power.

Horse-power measured in this way is known as brake

RELATION BETWEEN HEAT AND WORK 111

horse-power (B.H.P.), and represents the net power the engine
develops l irrespective of any frictional or other losses in the
engine itself.

The Relation between Heat and Work. — We have seen
that a heat engine takes in a certain amount of heat and that
some heat is rejected ; the amount rejected is found to be
iess than that taken in, so that when all the heat is accounted
for some heat disappears in the process. The amount which
disappears increases as the work done by the engine increases.
Thus in a heat engine heat disappears and work is obtained.
On the other hand, as we saw in Chapter I, when work is
done against friction heat is produced. So that heat can be
obtained from work and work from heat. Moreover it has
been established by the experiments of Joule (begun in 1840
and extended over many years) and others that there is a
definite rate of exchange ; if a certain amount of work is used
up a definite amount of heat is produced and vice versa.
The amount of work required to produce a unit of heat is
spoken of SLstheMechanicalEquivalentoi heat (denoted by J) ;
this is also the amount of work which is produced when one
unit of heat is completely used up in producing work.

Determination of the Mechanical Equivalent. — When
work is done against friction, heat is produced as described
in Chapter I. Thus, in determining the brake horse-power
of an engine the brake and the wheel become heated.2
If we can measure the total amount of heat produced at the
same time as the work done is determined we can at once
obtain the mechanical equivalent. This is the basis of a
very convenient laboratory method due to Callendar.

The general idea is to have a hollow brass drum mounted
on a shaft. Around the drum passes a silk belt, with a weight

1 The brake should be adjusted so that the engine runs at its normal
speed ; in practice it is usual to have a second spring balance in place
of the weight and some means of tightening the belt until the friction
is a suitable amount.

2 For this reason the method cannot be adopted for engines of large
power, since the amo\mt of heat developed is unmanageable, and other
methods of determining B.H.P. have to be used.

112

HEAT

suspended at one end and a spring balance at the other.
The number of revolutions of the shaft can be read off on a cyclo-
meter geared to it. Thus the work done can be determined
from the number of revolutions, the circumference of the
drum, and the difference between the forces at the two ends

FIG. 51.

of the belt. A known weight of water is placed in the drum,
and a thermometer bent into the shape shown in Fig. 51
enables the temperature to be determined. Thus as the drum
is heated by friction the water is heated also, and the amount
of heat developed is obtained from the weight of water, the
water equivalent of the drum and the rise in temperature.
Since the drum rotates while the water does not, every part

RELATION BETWEEN HEAT AND WORK 113

of the drum (except the part just near the axis) is constantly
being brought in contact with the water. Thus we can
measure the work done against friction and the heat produced
by the friction, and so we have the amount of heat produced
by this amount of work.

The apparatus is capable of giving extraordinarily consistent
and accurate results, but in order that this may be accom-
plished care is necessary in the measurement of the heat
developed, and the cooling correction (page 96) becomes
important. The writer has modified the procedure recom-
mended by the makers, and, as a number of interesting
points arise, the operations are described in some detail.

Fig. 51 shows the general arrangement. The silk belt
passes 1J times round the drum and is provided at each end
with a short brass bar. To one of these is hooked a fairly
heavy weight, while the other is not attached to the spring
balance but a frame of known weight (to which other small
known weights can be added) is hooked l on to it ; the spring
balance helps to support this frame. Thus the force on the
end of the belt is given by weight of frame and small weights —
spring balance reading. There is also attached to this end
of the belt a short piece of elastic provided with a hook ;
this hangs freely during the first part of the operation but
is used in the cooling correction.

The drum is driven, through reduction gearing, by an
electric motor, which should be provided with a controlling
resistance ; a suitable speed for the drum is about 120 r.p.m.

Two hundred and fifty gms. of water are put into the drum
by means of a pipette ; care should be taken not to get any
water on the outside. It is desirable to start with the water
at about atmospheric temperature. The motor is started
and the speed suitably regulated ; the large weight should
" float " between its stops.2 At each minute readings of the

1 As supplied by the maker, the frame is screwed to the bar.

* A preliminary adjustment is made by altering the point of attach-
ment of the spring balance to the frame ; further adjustment can be
made while running (if necessary) by adding or taking away on* or
more of the small weights, but see note, page 118.

1

114 HEAT

temperature, revolution counter and spring balance are taken :
after about eight or ten minutes the motor is stopped, readings
are taken, the heavy weight and the frame are unhooked
from the ends of the belt. The two ends are then connected
by the piece of elastic with the hook so that the belt con-
tinues to encircle the drum but now will revolve with it
freely, the motor is started up again and the speed adjusted
until it is about the same as before. All this takes about
half a minute. The temperature is then read every minute
with the drum still revolving, but no work being done by the
belt. This gives the rate of cooling with the conditions as
nearly as practicable what they were during the heating ;
the operation is continued for six or seven minutes to make
sure that regular cooling has set in.

The figures on p. 115 taken from an experiment will be
used to illustrate the method. Columns 1, 2, 5 and 6 give
the observed readings ; 3 and 4 are calculated as explained
later.

During the heating, the friction takes place at the outside
of the drum. As the drum is a good conductor and its sur-
face is constantly in contact with the water, most of the heat
developed goes to raise the temperature of the drum and its
contents and not much passes outwards through the silk ;
the mass of the latter is also very small compared with that
of the drum and contents. The temperature of the drum
must be slightly above that of the water, and when the motor
stops the temperature indicated by the thermometer does not
immediately fall, as some of the heat developed has not had
time to pass to the water.

Measurement of the Heat. Cooling Correction. — As
soon as the drum and its contents are at a higher temperature
than the surroundings, they begin to lose heat by conduction,
convection and radiation. This heat is supplied by the work
done and must be added to that determined by the rise in
temperature. If the temperature of a hot object is only a
few degrees above that of its surroundings it is fairly near
the truth to assume that the rate of loss of heat is propor-

Room temperature 13° C.

I !

I
Time

in ins. )

Temp.

Mean Temp, i
during the
Interval.

Rate of fall
of Temp, from
Graph.

Revn.
Counter.

Spring
Balance
(Ib. wt.)

Immedi-
ately after

0

13-4 ^

starting.

^>

14-4

•05

12,087

•55

1

15-4 ^

12,217

•55

^>

16-35

•15

2

17-3 >

*

12,347

•55

^>

18-25

•25

3

19-2

12,483

•55

^>

20-05

•3

4

20-9

12,620

•5

\>

21-75

•4

5

22-6 *

12,753

-5

6

24-2 <

23-4

•45

12,888

•45

7

25-8 ^

25-0

•55

13,026

•45

26-5

•6

8

2-75

13,161

Motor
stopped.

Weights
unhooked.

Motor
started.

81

27-2

about 27-1

•65

Fall per

3-40

9

26-8

minute.

^>

•6°

10

26-2

^>

•6

11

25-6 *

^>

•5

12

25-1 [

*^>

•5

13

24-6 *

^>

•5

14

24-1 '

^

•4

15

23-7 *

\

•4

16

23-3 ^

Load at one end of belt = 4,275 gms. wt.

Load at other end of belt = (473 gms. wt. — spring balance).
Mass of drum = 421 gms. Specific Heat of brass « *092.

w.e. of dram ^ 421 x '092 = 39 gms. wt.

115

116

HEAT

tional to the difference in temperature between the object
and its surroundings. This law, known as Newton's Law of
cooling, does not hold for large differences of temperature
(see page 237), but may be regarded as sufficiently accurate
for such cases as the present. To determine its amount we
take the observations after the cooling conditions have settled
down.1 From the 9th to the llth minute the rate of fall of
temperature is -6° C. per minute ; the mean temperature
during this interval is 26-2° C. Plot a graph in which abscissae
represent mean temperatures of the water, and ordinates the
rate of fall of temperature in degrees per minute.

22 20 18

Mean Temperature.

FIG. 52.

13, 12

The point P represents the rate -6° per minute for a mean
temperature 26-2° C. Join this point by a straight line to
13° C., the room temperature. This straight line gives us
the rate of fall of temperature corresponding to any given
mean temperature. (As a check, Q represents the rate,
•5° C. per minute corresponding to 24-8°, got from the readings
from the llth to the 14th minute.)

From this graph we get the rate of cooling corresponding
to the mean temperatures shown in column 3 of the table.
These values are shown in column 4. During the first eight
minutes, the total loss of heat is that given out by the drum

1 When the temperature is falling, the drum must be slightly cooler
than the water ; it is assumed that this condition has set in after the
9th minute, since the rate of cooling is gradually diminishing. From
the 8th to the 9th minute the fall of temperature is less than that which
is observed later, when the mean temperature is lower ; this must
mean that the fall in temperature of the water does not indicate the
whole of the heat loss.

RELATION BETWEEN HEAT AND WORK 117

and contents falling through the sum of all the separate
amounts, i.e. 2-75° ; during the 9th the loss of heat is that
corresponding to a fall of -65° C. If none of this heat had
been lost the temperature at the end of the 9th minute would
have been 26-8 + -65 + 2-75 = 30-2° C.1 Thus, if no loss
had occurred, the final temperature of the water would have
been 30-2° C.

We can now calculate the value of J.

To get the heat produced we have 250 gms. of water -f the
drum were heated from 134° C. to 30-2° C. Taking the water
equivalent of the drum as 39 gms., this corresponds to 289 gms.

289

of water or - - Ib. of water.
454

900

Heat produced therefore = j 7 X 16-8 T.U.

454

To get the work done —
Load on one end of belt = 4,275 gms. wt.
Load on other end = (473 — mean spring balance

reading)

= (473 — -51 X 454)
= 242 gms. wt.
Force of friction = 4,275 — 242 = 4,033 gms. wt.

.
454

Diameter of drum = 6 inches = '5 feet.

Work done per revolution == ~~ X n X -5 foot-pounds

•i.o4

Total revolutions = 13,161-12,087 = 1,074.

Total workdone = 4033 x ,5 x_Lg74

4'033 X ^'^ uce gg x 16-8 T.U.

= 1,401 foot-pounds per T.U.

1 Observe that during the 9th minute the fall in temperature recorded
is less than that corresponding to the mean temperature, which means
that some of the heat generated by friction is still coming in. After
the 9th minute, if we add on the cooling correction the temperature
would remain constant since no more heat is coming in.

118 HEAT

Note on the Above Experiment. — It is sometimes pointed
out that the cooling correction can be avoided for practical
purposes by starting with the water at a temperature below
that of the room and stopping when the temperature is about
the same amount above that of the room. This arrangement
limits the range of temperature, since if the water starts at
more than 3° or 4° below room temperature one is apt to get
a deposit of dew on the drum, in which case the friction is
very irregular. The writer prefers to obtain a larger range
and apply a cooling correction. But the cooling correction
is very unreliable if the drum is at rest during the observa-
tions ; the water should be continually brought into contact
with all parts of the drum, if the readings of the thermometer
are to be taken as representing the mean temperature of the
drum and its contents. Emission of heat takes place from
all parts of the drum, and if the water remains in one place
and is not stirred the temperature will vary at different
parts. With the drum rotating, the writer finds that the
cooling is very regular ; with the drum at rest it is very
irregular. Even as it is, the drum being hotter than the
water during heating, the loss of heat must be somewhat
greater than that deduced from the cooling curve ; the design
of the apparatus is such that this difference is small. The
tendency is therefore for J to come out rather above than
below the accepted value. One other point may be noticed.
The instrument is usually supplied with a spring balance
reading in grammes ; the writer substitutes a rather stiffer
one reading to TV lb. By this means the minor variations
in the friction are much less troublesome ; no adjustment is
necessary during the running. With the more sensitive
spring greater care is necessary in attending to the polish
of the friction surface than is convenient in an ordinary
laboratory. One- tenth lb. is only about 1 per cent, of
4,275 gms., so that the measurement of the work is quite
as accurate as the determination of temperature is likely
to be ; for the same reason it is legitimate to take a mean
value for the spring balance reading as described, instead

RELATION BETWEEN HEAT AND WORK 119

of getting the separate values of the work done in each minute.
By adopting the procedure described, the writer finds that
without any trouble or previous preparation the result obtained
can be trusted to be within about 2 per cent, of the accepted
value, i.e. somewhere between 1,400 and 1,430 foot-pounds
per T.U., and although the apparatus may be capable of
greater accuracy, comparatively few people are likely to get
the temperature measurement better than within 2 per cent.
without more elaborate precautions.

The Thermal Efficiency of an Engine. — Since a unit
of heat is equivalent to a definite amount of work, and we
can find the calorific value of any kind of fuel, we are able to
calculate the amount of work which should be produced by
burning a known weight of fuel if all the heat were converted
to work. Thus, if we know the work which an engine can
do in a given time (as determined by the B.H.P.) and the
amount of fuel it uses in that time, we can find out whether
the engine is wasteful or not.

The thermal efficiency of an engine is defined as —

Work done by the engine in a given time
Work equivalent to the heat value of the fuel used in that time.

To take an example. A good steam engine of large size
is found to take about 1-3 Ib. of coal per hour for each horse-
power developed. Taking the calorific value of coal as about
8,000 T.U. per Ib. we have—

1-3 Ib. of coal give 1-3 x 8,000 T.U. = 10,400 T.U.

But since 1 T.U. corresponds to 1,400 foot-pounds, 10,400
T.U. corresponds to 10,400 X 1,400 = 14,560,000 foot-pounds.
But the work actually got from this is 1 horse-power for
1 hour

= 33,000 X 60 = 1,980,000 foot-pounds.

mi ~ . Work actually done

Thermal efficiency = — — =-£ -- — -

Work ideally possible

In practice the thermal efficiency of heat engines is low ;
13 per cent, is not a bad value for large steam engines.

120 HEAT

Internal combustion engines may have a higher value under
certain conditions ; some Diesel engines have reached 38 per
cent., while the Still engine (a combined steam and explosion
engine) is claimed to have given 42 per cent.

This low efficiency is to some extent due to mechanical
imperfections, but more to the fact which we have already
noticed that heat engines do not make use of all the heat
taken in, but return a large amount to the atmosphere or
condenser and this heat is of no use to the engineer. As we
shall see in Chapter XXI, there is no hope of avoiding some
rejection of heat, so that no improvement of design is likely
to bring the efficiency of any heat engine anywhere near
100 per cent.

The Nature of Heat. Development of Modern Ideas. —

Down to about the end of the eighteenth century, the prevailing
ideas about the nature of heat were summed up in the caloric
theory ; according to this theory heat was a kind of invisible
fluid J called " caloric " which filled the spaces between the
particles of matter. On this theory, the fact that percussion or
compression gave rise to " sensible heat " was accounted for
readily enough ; the caloric was supposed to be squeezed out,
like water out of a sponge. It was not quite so easy to account
for the production of heat by friction, but it was considered that
the small particles of the material which were rubbed off had a
smaller capacity for heat than the original solid, and so some of
the caloric made its appearance. There seems to have been no
attempt to determine by experiment whether the capacity for
heat of the small particles was really less than that of the
solid.

In 1798. however, experiments were made by Benjamin
Thomson, Count Rumford. He had been struck by the very
great development of heat during the boring of brass cannon in
which he was engaged at Munich. He carried out experiments
on a small hollow cylinder of gun metal, some 10 inches long and

1 It was generally regarded as being imponderable, but as late as
1785 a paper is to be found in the Phil. Trans, in which the author
describes experiments undertaken to find out whether it was possible
to detect any change in weight of a body to which caloric is supplied
and he concluded that he had detected a change.

RELATION BETWEEN HEAT AND WORK 121

8 inches diameter. He used a blunt borer, as he wanted to see
whether less heat would be developed in consequence of the
conversion of a smaller amount of material into metallic chips
and dust. He found that a very large amount of heat was
developed, in spite of the fact that the amount of metal removed
was very small. Furthermore, in order to see if the exclusion
of the atmosphere had any effect, he placed the cylinder in a box
filled with water ; after a time (2| hours) the water boiled.
There was therefore no question of heat being abstracted from
the atmosphere.

He pointed out that a very large amount of heat was given off
continuously in all directions ; it was difficult to imagine that
this could come as the result merely of a change in the capacity
for heat of the insignificant amount of material removed. More-
over he showed by experiment that the capacity for heat of the
metallic chips was practically the same as that of the solid. He
lays great stress on the fact that the supply of heat is apparently
inexhaustible ; as long as the borer is worked there is no sign of
any exhaustion. He concludes : " It is hardly necessary to add
that anything which any insulated body or system of bodies can
continue to furnish without limitation cannot possibly be a
material substance ; and it appears to me to be extremely diffi,
cult, if not quite impossible, to form any distinct idea of anything
capable of being excited and communicated in the manner the
heat was excited and communicated in these experiments except
it be MOTION." J

Experiments performed by Humphry Davy were however even
more strikingly at variance with the caloric theory. Davy
arranged that two pieces of ice should be rubbed together, and in
the later experiments this was done by clockwork, the whole
apparatus being under the receiver of an air pump. The ice was
found to be melted at the surface where the rubbing took place
and not elsewhere. Now in order to turn ice to water, heat is
required (see Chapter IX), while the capacity for heat of water is
greater than that of ice. The heat, therefore, could not have
come out of the ice, as the result of its change to water.

It is difficult to conceive of any single experiment which is
more contrary to the reasoning of the calorists, but the fact
remains that they were not convinced, and even Davy himself

1Rumford, Phil. Trans., 1798.

122

HEAT

appears afterwards to have had doubts of which he did not rtfl
himself for some time.

The experiments of Davy and of Rumford mark the beginning
of the modern idea of heat, but the real foundation was laid by
the experiments of Dr. Joule of Manchester, begun in 1840 and
continued for many years. Rumford's account of his experi-
ment makes it fairly clear that he had a general idea that the
amount of heat depended on the work done, but Joule set out to
determine the matter with accuracy ; he arranged experiments
in which the work done and the heat produced were measured.
In one set of experiments water was churned by a revolving
paddle ; the paddle was driven by means of weights which
descended a known distance.

Fig. 63 shows a rough idea of the arrangement. W, W are two

FIG. 53.

weights, which by their descent cause the vertical shaft C to
rotate and drive the paddle in the calorimeter which contains a
known weight of water. In order that the water might not be

RELATION BETWEEN HEAT AND WORK 123

merely carried round with the paddle and so give rise to very
little friction, the inside of the calorimeter was provided with
fixed partitions with openings through which passed the arms of
the paddle, as shown in the section in Fig. 53. When the
weights reached the ground the paddle shaft was disconnected
from the roller by the screw D (Fig. 53) and the weights quickly
wound up again. The shaft was connected again and the weights
released. In this way some twenty descents of the weights were
carried out, while the rise in temperature of the water was
determined by a thermometer.

The work done was determined by the total distance the
weights descended, a correction being applied for the friction of
the bearings and for the fact that the weights reached the ground
with a certain speed and so possessed kinetic energy.

The heat produced was determined from the mass of water
+ water equivalent of calorimeter and paddles and the rise in
temperature. Corrections were applied for loss of heat to the
surroundings.

Joule carried out experiments in which the heat was generated
in other ways ; he used liquids other than water and also the
friction between solids, compression of air, etc. In all these
experiments he found that the amount of work required to pro-
duce a unit of heat came out as about the same ; he regarded the
method of churning water as most reliable, and devoted a great
deal of time to that method. Joule gave as the value for the
Mechanical Equivalent 772 foot-pounds * for one B.Th.U., i.e.
the amount of heat necessary to raise the temperature of 1 Ib. of
water 1° F. The temperature was measured by a mercury in
glass thermometer and was not reduced to the air scale. When
this correction is applied to Joule's results, the value comes out
as about 778 foot-pounds per B.Th.U.

About 1857 Him carried out experiments with a steam engine
in which he measured the heat taken in from the boiler and the
heat given out to the condenser. After allowing for heat lost in
the cylinder by conduction and radiation he found that heat had
disappeared, and the amount which disappeared was very much
greater when the engine was driving machinery than when it was
running light. Deducting the heat returned to the condenser and
the heat lost by radiation and conduction from that taken into

1 Taking the weight of 1 Ib. in the latitude of Manchester.

124 HEAT

the cylinder from the boiler the amount of heat used up in doing
work was determined, and thus a value for the mechanical equi-
valent was obtained. The result obtained by Him in this way
was in very fair agreement with that obtained by Joule and others
for the inverse operation. The difficulties of the experiment
were great, and a very accurate result was not to be expected ;
it is however of importance as a direct determination of the
amount of work produced by a unit of heat.

These experiments are the foundation of the modern view that
heat is a form of energy. This is merely a particular case of a
more general Principle, known as the Principle of the Conserva-
tion of Energy. This Principle states that energy cannot be
created or destroyed, it can merely be changed from one form to
another. Energy, the capacity to do work, can exist in many
forms, thus a weight at the top of a cliff possesses energy due to
its position, for it can do work by being allowed to descend. A
projectile possesses energy due to its speed, for it can do work in
the course of being brought to rest. Energy may exist as elec-
trical energy or in other forms. Coal possesses chemical energy ;
if burnt it gives out heat. Thus we might burn coal in the
furnace of a steam engine and obtain mechanical work ; the
engine might drive a dynamo giving electrical energy, and this
might be used to drive a motor.

In all these eases there is no loss of energy, only a change of
form. It is true that a five horse-power engine would not drive
a dynamo capable of giving enough energy to drive a five horse-
power motor ; but in this case all the missing energy can be
accounted for in the form of heat developed by friction and in
other ways. The Principle of the Conservation of Energy then
states that the total amount of energy in an isolated system is
constant.

The whole of modern physical science is built upon this Prin-
ciple, and no experimental result has been found contrary to it ;
predictions based on it are found to be true.

It is of great importance in Physical Science that the value 01
the Mechanical Equivalent should be known with accuracy, and
many experiments have been made to determine it. A complete
account of these experiments would take up far too much space ;
we may summarize them as follows : —

Direct methods, in which water is churned, the mechanical
work and the heat produced being measured. In all of these the

RELATION BETWEEN HEAT AND WORK 125

method of driving the paddle by descending weights is abandoned,
and the work done is measured by some form of friction balance.
This method is originally due to Hirn. It may be illustrated by
a brief description of Joule's apparatus in his second set of
experiments as shown in Fig. 54.

The paddle was mounted on a vertical shaft A as before, but
it was driven by hand wheels instead of weights. The calori-
meter C did not stand on a fixed support, but was suspended on
a bearing so that it could rotate freely about A. When the paddle

W

FIG. 54.

was revolving the calorimeter tended to rotate with it. A couple
tending to prevent rotation was applied to the calorimeter by two
cords which left the calorimeter at opposite ends of a diameter
and passed over two pulleys P, P ; to the ends of these cords
equal weights W, W were attached. The paddle was driven at
such a speed that the weights were raised from the floor and kept
suspended in one position. Thus the couple due to friction in
the calorimeter just balanced that due to the weights.

If r is the radius of the calorimeter where the cords were
attached, this couple is 2rW when W includes the scale pan.

The work done against friction when the paddle is turned

126 HEAT

through an angle 0 is therefore 2rW0 ; and in n revolutions it
would be 2nn x 2rW = 4?mrW.

The number of revolutions of the shaft was determined by a
revolution counter geared to it.

The point of this arrangement is that the work done in the
calorimeter is measured, and so no corrections due to friction of
driving cords and bearings come in ; there are also other advan-
tages. The calorimeter was supported on a hydraulic support
which did not hinder rotation and avoided trouble due to the
weight of the calorimeter on the bearing and any want of centring.

Experiments of this type were performed by Joule x and Row-
land,2 while others of a similar principle were those of Micelescu3
and Reynolds and Moorby * ; in these latter experiments the heat
measurement was different, in that water flowed through the
calorimeter and the rate of flow was adjusted until a steady state
was reached in which the temperature of the outflowing and
inflowing water was constant ; the heat produced in a given
time is thus measured by the mass of water which has gone
through in the time x the difference in temperature of the out
flowing and inflowing water.

The advantage of this is that the water equivalent of the
apparatus does not come in, and the corrections for loss of heat
are much easier. Reynolds and Moorby used large powers (100
H.P. steam engine) and arranged that the water should come in
at a temperature not much above 0° C. and come out nearly at
100° C., the idea being to avoid uncertainties of the scale of
temperature.

Electrical Methods. — Of the indirect methods of deter-
mining J, the electrical are the most important. The general
principle is to send a known current through a wire of fairly high
resistance immersed in liquid in some form of calorimeter, so as
to measure the heat produced in a given time. If C is the current
in amperes and E the difference of potential in volts between the
ends of the resistance, we have from the definition of the electrical
units that work is being done at the rate of EC joules 5 per
second.

Trans., 1878.
1 Proc. Amer. Acad. Arts and Sciences, 1879-80

* Ann. de Chimie et Physique, 1892.

* Phil Trans., 1897.

* 1 joule = 107 ergg,

RELATION BETWEEN HEAT AND WORK 127

The heat produced in a given time is measured in the usual
way. Experiments of this type were performed by Schuster and
Gannon * and by Griffiths * ; while Callendar and Barnes •
performed a series of experiments in which the heat was measured
by having a flow of water through the apparatus and arranging
that a steady state should be attained as in the experiments of
Micelescu and Reynolds and Moorby. A steady current of liquid
flowed down a narrow tube, along the axis of which is a wire
carrying a current C. A pair of differential platinum thermo-
meters, Pj, P2, gave the difference in temperature of the inflowing
and outflowing water. The potential difference at the ends of
the conductor was measured on a potentiometer in terms of a

Fio. 55.

standard cell, and the current was measured by including a
known standard resistance in the circuit and measuring the P.D
at its ends on the potentiometer.

Variation of the Specific Heat of Water. — As soon as
accurate methods of measuring J were available it was seen that
the amount of work required to raise the temperature of a given
weight of water through 1° depended on the part of the scale of
temperature at which the 1° was taken. Before this time
experiments of the ordinary method of mixture type had shown
variations, but the results were uncertain ; the mechanical
equivalent method is much more reliable. The most trustworthy
results are those of Dr. Barnes with the apparatus described in
the last paragraph. His results show that the Specific Heat of
water is a minimum at about 37*5° C.

The Unit of Heat. — A calory is denned as the amount of
heat required to raise the temperature of 1 gm. of water 1° C. It

1 Phil Trans., 1895.
*Phil Trans., 1893.
• Phil. Trans., 1902.

128 HEAT

is evident that this definition has no strict meaning unless we
specify at what part of the scale the 1° C. is to be taken.

The following units are recognized : —

The 15° calory = heat required to raise temperature of 1 gm.

of water from 14-5° to 15-5°.
20° calory = 19-5° to 20-5°.

The mean cal. = Ti5 of heat required to raise 1 gm. of water
from 0° to 100° C.

The mean and 15° calory are very nearly equal and equivalent
to 4*184 joules ; the 20° is about 1 in 1,000 less and equivalent
to 4-180 joules.

CHAPTER IX
CHANGE OF STATE— SOLID TO LIQUID

If we take a solid substance and heat it, in general the
temperature rises and a certain amount of expansion may take
place. If we go on, a stage is reached when the solid begins
to melt and turn to a liquid.1 At this stage certain important
effects occur.

In the first place it is found that, in general, there is a
very perceptible change in volume. In the majority of cases
the liquid occupies a larger volume than the solid. This may
be illustrated in quite a striking way by melting some paraffin
wax in a large test tube and then allowing it to cool. The
outside cools first and the result is that when the whole mass
is solid, the surface instead of being flat has a deep hollow
in it, and there may even be a distinct cavity in the centre.
It may be noticed that when pieces of the solid are dropped
into the molten wax the pieces sink, indicating that the solid
has a greater density than the liquid.

In the case of a few substances, of which water is a striking
example, the effect is in the opposite direction ; when water
turns to ice the volume of the ice is decidedly greater than
that of the water. Roughly speaking, 11 c.c. of water form
about 12 c.c. of ice. This increase in volume may be shown
in the following simple manner (Fig. 56).

A large test tube is half filled with water ; paraffin oil or

1 This is thb. normal course : exceptions are furnished by substances
which decompose before they reach the melting point, and a few sub-
stances turn to vapour without passing through the liquid phase, and
the vapour if cooled passes back direct to the solid state. Arsenic is an
example.

129 K

130

HEAT

:

turpentine is poured in on the water until the tube is full.
The tube is then closed by a cork through which passes a
length of quill tubing open at each end ; care must be taken
to see that there are no bubbles of air under the cork, and the
oil should rise a little way up the quill tube. The test tube
is then placed in a freezing mixture of ice and salt. Nothing
much appears to happen for a little while, although the oil
may sink slightly ; but after a time the oil begins to rise
quite quickly and ultimately may flow out of the
upper end of the quill tube. If the test tube is
now taken out. of the freezing mixture the water
will be seen to be wholly or partially turned to
ice. The paraffin oil or turpentine does not solidify
at the temperature of the freezing mixture.

A more striking illustration is to take a small,
thick-walled cast-iron bottle, fitted with a screw
stopper. The bottle is completely filled with
water and its stopper screwed home. It is then
completely immersed in a freezing mixture of ice
and salt contained in a metal dish ; a duster should
be placed over the dish. After some considerable
time the bottle will burst with a loud crack ; it
usually splits cleanly, but the writer has known a
large piece to be driven up to the ceiling, and there-
fore recommends the duster.

In winter this behaviour is sometimes emulated
FIG. 56. by" the water-pipes of a house, to the discomfiture
of the occupants. The water in the pipes freezes
and splits the pipe by its expansion ; the effect is not
usually noticed at the time, but as soon as the thaw comes
the water escaping from the rent draws attention to the
phenomenon. This trouble is much less likely to happen
if the pipes are elliptical in section, since the area of an
ellipse is less than that of a circle of the same perimeter
and so there is the possibility of some expansion.

The action of frosts in breaking up and loosening the soil
— an impo^ant effect from the point of view of agriculture —

SOLID TO LIQUID 131

is another example of this property of water. The water
which has percolated into the soil expands when it freezes
and the result is that the soil is much broken up ; this is
very evident as soon as the thaw comes. The weathering
of rocks at high altitudes is also largely due to this action ;
water percolating into cracks by day freezes at night, splitting
off pieces of the rock. In many cases the ice holds the piece
in position until the sun gets up and melts the ice, when falls
of rock occur ; on the other hand, big masses too large to be
bound by the ice are split off and fall at night ; and Mr.
Whymper noticed that on each of seven nights which he
spent on the south-west ridge of the Matterhorn the greatest
rock falls occurred in the night between midnight and
dawn, this period being the coldest part of the twenty-four
hours.

The Latent Heat of Fusion. — Another phenomenon
connected with change of state is that when a solid turns
to a liquid heat must be absorbed by it. This may be
illustrated by the experiment shown in Fig. 57.

A vessel of thin copper is supported from the ring of a
retort stand. A stirring gear, consisting of a vertical shaft
with a piece of wire gauze soldered to it and bent in the shape
of a propeller is provided ; the propeller must be near the
bottom of the vessel. A thermometer is placed in the posi-
tion shown. The vessel is filled with small pieces of ice, care
being taken to put in as much as possible. The temperature
is noted, and the apparatus is lifted by the stand and placed
so that the copper vessel is in the steam from a large beaker
of boiling water. The thermometer is read every half -minute.
A. small motor is connected to the stirrer by a string belt.
At first the stirrer cannot turn, but when a fair amount of ice
is melted, the remainder floats and the propeller can turn
in the water at the bottom.

Readings of the temperature are continued until all the
ice is melted, and for some time after ; as long as any ice is
present it can be heard rattling against the vessel and ther-
mometer, and so it is easy to tell when it has all gone. The

132

HEAT

FIG. 57.

readings (p. 133) obtained in an experiment illustrate what
happens.

The ice had all melted just after the 8th minute. The
propeller started at the 4th minute. The results are plotted
on the graph shown in Fig. 58.

Now the vessel was receiving heat from the steam all the
time, but until the ice has all been melted the temperature
hardly rises appreciably ; it does not do so at all while there

SOLID TO LIQUID

133

Time.

Temp.

Time.

Temp.

Time.

Temp.

mins.

°C

mins.

°C

mins.

°C

0

0

4

0

8

3-5

i

0

H

•5

s|

7-5

1

0

5"

1-0

9

12

1J

0

5£

•5

91

17

2

0

6

•5

10

22

2*

0

6£

1-0

10i

27

3

0

7"

1-5

11

32

3£

0

n

2

Hi

36

12

40

is a, fair amount of ice present, and the small rise of 3°C.
at the end is due to the fact that with a very small amount
of ice the stirring cannot keep all the water at the temperature

35

15

0^

Ice

O1 23456789 fOfl
Time.

FIG. 68.

134

HEAT

of that immediately in contact with ice.1 Once the ice is
melted the temperature rises at the rate of about 5° C. each
half -minute ; when the temperature of the water approaches
that of the steam, the rate of supply of heat diminishes, as
shown by the last two readings.

A second experiment, with an equal mass of water cooled
to as low a temperature as possible, but with no ice, gave the
following figures : —

Time.

Temp.

Time.

Temp.

mina.

°C

mins.

°c

0

4

2

22-5

1

8

2*

27-5

1

13

3

32

1*

18

3*

35-5

4

39-5

The early readings of this again correspond to a rate of
rise of temperature of just under 5° C. per half-minute.

Thus, while ice is melting heat is being supplied to it, but
the resulting water is at much the same temperature ; in
our case we had to supply this heat for just over eight minutes
to melt all the ice. Calling the mass of ice x gms. (this being
also the mass of the resulting water), the heat was supplied
at the rate of about 5x calories per half-minute, and therefore
in eight minutes 8Qx calories were supplied. Thus to melt
x gms. of ice requires something like 80a; calories (strictly
speaking, the SOx calories also raised the temperature of the
water 3°, so that llx calories were used in melting the ice ;
since, however, we have neglected the water equivalent of
the vessel, we shall not be far out in calling the heat required
to melt the ice SOx calories, and the value is only intended
to be rough), or 80 calories per gm.

1 Since the heat is being communicated from the outside through
water to the ice, part of the water must be at a slightly higher tem-
perature than the rest, otherwise the transmission of heat would be
infinitely slow. See Chapter on conduction of heat.

SOLID TO LIQUID 135

This behaviour of ice is characteristic of solids ; in order
to turn a solid into a liquid, heat must be supplied although
the resulting liquid is at the same temperature. Since this
heat seems to disappear (i.e. there is no corresponding rise
in temperature), it is spoken of as latent l (meaning hidden).
In reality it is a case of transformation of energy, and when
the liquid solidifies the latent heat is given out.

Definition of Latent Heat of Fusion.— The latent heat
of fusion of a substance is the amount of heat required to
turn unit mass of the substance from solid to liquid without
change of temperature.

Determination of the Latent Heat of Fusion of Ice. —
The method just described is only intended to give a rough
value ; it happens to have come out to a value very near
the truth, but could not be relied on to give very accurate
figures, and is only intended as an illustration.

The determination is usually carried out by the method
of mixtures as follows : —

A calorimeter is weighed, and is then about two-thirds
filled with water and weighed again ; this gives the weight
of the water. The temperature of the water should be about
6° or 7° above room temperature. The calorimeter is placed
on corks and the temperature of the water carefully deter-
mined. Some ice, broken up into fairly small pieces, has
previously been got ready. Ice is added to the water in the
calorimeter one piece at a time ; the piece of ice should be
roughly dried on a cloth or blotting-paper before adding, and
the water thoroughly stirred. As soon as the piece of ice
is melted another should be added ; the process should be
continued until the temperature of the water is a few degrees
below the room temperature.

This temperature is read carefully and the calorimeter and
its contents are then weighed ; by this means the weight of

1 This name is due to Black, who first noticed the effect. The experi-
ment described above is similar in principle to an experiment performed
by Black, but is much more rapid.

136 HEAT

ice added is determined. The following figures obtained in
an experiment will illustrate the procedure.

Room temperature = 17° C.

Weight of copper cal. . . . . . = 192 gms.

Weight of cal. -f water . . . . . = 649 gms.

. ' . weight of water . . . = 457 gms.

First temperature of water . . . . = 26-5° C.

Final temperature of water . . . . = 13-5° C.

Weight of cal. after adding ice . . . = 714 gms.

. ' . weight of ice . . . . = 65 gms.

Water equivalent of calorimeter . . . = 19 gms.

(about).

Considering the water originally in the calorimeter 457 gms.
of water + the calorimeter have cooled from 26-5° C. to
13-5° C.
. • . they have given out —

(457 + 19) X 13 calories

= 6,188 calories.
Now this heat has been employed in two operations: —

(1) Turning 65 gms. of ice at 0° C. to water at 0° C.

(2) Raising the temperature of this 65 gms. of water which
comes from the ice from 0° C. to 13-5° C.

But this last operation requires 65 x 13-5 calories

= 877 calories.

. • . 6,188 calories have been used in melting 65 gms. of
ice without change of temperature and supplying a further
877 calories.

/.To melt 65 gms. of ice requires . 5,311 calories

„ 1 gm. „ ,, 82 calories

(to nearest whole number).

So that the latent heat of fusion of ice is 82 calories per gm.

The above experiment can be trusted to about 4 per cent.

The most accurate determinations give a value rather

under 80 calories per gm. ; that is to say, nearer 80 than 79.

The Melting-point of a Solid. — If we take a piece of ice
at a low temperature and gradually heat it, we reach a tem-
perature at which it begins to melt. We have here an abrupt
change ; there is no gradual softening of the ice ; it is either
hard ice or liquid water, and the change takes place at a

SOLID TO LIQUID 137

perfectly definite temperature (but see page 142 on the
influence of pressure), and while the change is going on the
temperature remains constant. As we have already seen,
change in volume occurs, and there is an absorption of heat
without rise of temperature ; moreover the specific heat of
ice is quite different from that of water.

A large number of solids behave in the same way and
melt at some definite temperature with an abrupt change
in volume and an absorption of heat. Some however, mix-
tures such as paraffin wax and amorphous substances, do not
have a perfectly definite melting-point ; the solid softens and
gradually passes into the liquid form, so that it is only possible
to give a range of temperature within which melting occurs,
and not a definite melting-point (see page 142).

In the case of solids with a definite melting-point, the
liquid if cooled will, under ordinary conditions, freeze or
solidify at the same temperature as that at which the solid
melted ; the latent heat is given out again and the tempera-
ture remains constant while the change is going on.

Super -cooled Liquids. — If water is cooled slowly and
very carefully it is possible to get it to a temperature well
below 0° C. without its freezing. If, however, it is then
shaken, or better, if a fragment of ice is dropped in, the water
at once freezes and the temperature rises to zero. The water
below 0° C. is said to be super-cooled, and all substances with
definite melting-points exhibit this phenomenon. It should
be noticed that the introduction of a fragment of some other
solid into a super-cooled liquid will not necessarily bring about
solidification ; the only certain way is to introduce a fragment
of its own solid. Thus, the only true test as to whether a
liquid is at its freezing-point is to see if it can remain in con-
tact with its solid for an indefinite period without change
occurring.

Although liquids can be super-cooled it does not seem
possible to heat a solid above its melting-point ; for this
reason, although melting-point and freezing-point of a sub-
stance are frequently used as alternative terms, it is usual

138

HEAT

to define 0° C. as the temperature at which pure ice melts,
rather than that at which pure water freezes, as there is then
no possibility of ambiguity.

Determination of the Melting-point of a Solid.

A method suitable for cases where only a small quantity
of the substance is available, and one which with skilled
observers gives good results is as follows.

A small capillary tube (made by drawing out quill tubing)
closed at the lower end contains some
of the solid. It is fastened to the
bulb of a thermometer by rubber bands
and the thermometer placed in a
beaker of water l as shown in Fig. 59,
so that the upper end of the capillary
is above the water. The beaker is
gently heated and the water well
stirred ; as soon as the solid is seen to
melt (which is noticed because it then
becomes more transparent) the tem-
perature is noted. The water is allowed
to cool and the temperature at which
the substance solidifies again is noted.
If the experiment is performed with
care this should not differ by more
than a degree or two from the first

temperature ; in this case the mean of the two tempera-
tures may be taken as the melting-point.

A more accurate method, but one which takes a much
longer time, is as follows : —

Determination of the Melting-point of a Solid by the
Method of Cooling. — This method makes use of the latent
heat of fusion. The apparatus described below is suitable
for a laboratory determination when the melting-point is not

1 This only applies to solids having a melting-point below 100° C.
Other liquids can be used if the melting-point is above 100° C., but the
method is not very suitable for such cases.

FIG. 59.

SOLID TO LIQUID

139

too high. The method can, however, be adapted to meet
the case of solids with high melting-points, such as metals
and alloys ; in that case the temperature measurement would
be done by a thermo-couple or some such means.

A quantity of the solid is placed in a test tube,1 and gently
heated until it is all melted ; it is then heated until its
temperature is about 20° or 30° above the melting-point.
(This temperature is estimated and care is required to avoid
heating the liquid to a temperature
above the range of the thermometer.)
The test tube is then placed inside a
larger tube, as shown in Fig. 60, a
thermometer is suspended with its bulb
in the liquid ; the thermometer bulb
must be covered by the liquid and
must not touch the bottom of the
tube. The amount of substance taken
should be enough to cover the bulb
easily ; a large excess means that the
experiment will take a very long time.
A stirrer is also placed in the liquid.

The hot liquid is then allowed to
cool, and the reading of the thermo-
meter is noted every minute ; con-
stant stirring being maintained. At
first the temperature falls fairly
rapidly, as the tube is losing heat to
its surroundings ; but as soon as the
liquid begins to solidify the latent heat
of fusion is given out and this balances
the heat lost to the surroundings. The result is that the

1 The size of the tube is chosen to give a suitable rate of cooling.
The smaller the tube the more rapidly will the temperature fall ; it
must, however, be large enough for a stirrer. In the experiment
described, the inner tube was 6 inches x £ inch and the outer 8
inches X 2 inches. An outer tube is essential to avoid disturbances
in the rate of cooling which would give a misleading appearance
to the result*.

FIG. 60.

140

HEAT

temperature remains constant. When all the liquid has
solidified there is no more latent heat to balance the loss
to the surroundings and the temperature then begins to
fall again. A graph is plotted, showing the variation of
temperature with time.

Fig. 61 shows the graph obtained with a specimen of phenol.

0 4 & 12 16 .20 24. 28 3Z 36 40 44- 48
Tune (Minutes)

FIG. 61.

It will be seen that the temperature falls rapidly until at the
8th minute it has dropped to 41° C. At this stage it remained
constant for twelve consecutive readings. At about the 20th
minute so much of the phenol had solidified that stirring was
no longer possible. The temperature then begins to fall ;
at first very gradually. The first gradual fall is probably
due to the impossibility of stirring ; at this stage most of the

SOLID TO LIQUID 141

substance is solid so that the lower part of the thermometer
bulb is in contact with solid, the outer part of which is
beginning to cool, while the upper part of the bulb is sur-
rounded by the last remaining liquid, still at 41° C. The
mean temperature of the bulb's surroundings is thus slightly
below 41° C. The fall of temperature soon becomes unmis-

t>u<
56
52
48

44

36
32

28
24
tn

\

\

\

\

\

\

\

p,

xraffin 1

Wax

-

\

>— »i

~— ,

*— t

^*s

>— 1

^"^

•^^

^^

S

s

Ss

^

^

^

S

t

s

V

S

8 12 16 26 24 28 32 36
Time (Minutes)

FIG. 62.

takable and we get the ordinary cooling curve of a hot solid.
From an inspection of the graph we can at once see that the
melting-point is 41° C. ; we have rapid cooling until 41° C.
is reached, then constant temperature for some time at 41° C.,
and then a further fall.

Such a graph is characteristic of a pure single chemical
substance (element or compound) ; there is one sharply-

142

HEAT

defined melting-point. In the case of substances which are
not single substances but are mixtures such as fats and waxes,
a different result is obtained. Fig. 62 shows the graph
obtained with paraffin wax in the same apparatus as in the
previous experiment. Here it will be seen that the tempera-
ture falls rapidly until it reaches 45° C. At this stage the
rate of fall of temperature abruptly diminishes, but the
thermometer readings do not remain constant for several
consecutive minutes. It is not a question of want of stirring,
because stirring was maintained until the 10th minute. There
is a slight but irregular fall of temperature until 42° C. is
reached. After falling below 42° C. the temperature continues
to fall rapidly and the curve then resembles the phenol curve.
Thus there is no definite melting-point for the paraffin wax ;
solidification is taking place over the range 45° C. to 42° C.
In some cases the graph can be seen to develop a number
of horizontal steps, corresponding to melting-points of different
constituents.

Effect of Pressure on the Melting-point. — The melting-
point of a solid is affected by pressure ; in the case of ice which
occupies a larger volume than its
liquid, increase of pressure lowers
the melting-point, while in the case
of solids like paraffin wax increase
of pressure raises the melting-
point.1 The effect is not large ; in
the case of ice, an increase of pres-
sure of one atmosphere lowers the
melting-point by about -0076° C.
This lowering of the melting-point
is well illustrated by the following
experiment (Fig. 63) : —

A large block of ice is placed on two
63. supports, and round the middle of it is

placed a loop of thin wire from which

1 This effect can be predicted from considerations of energy, se*>
Chapter XXII, p. 333.

SOLID TO LIQUID 143

heavy weights are suspended. Since the area of the wire in
contact with the ice is small, the weight produces a large
pressure per unit area under the wire. The ice under the
wire melts, and the wire sinks in ; the water formed, flowing
round the wire, is released from pressure and freezes again.
This goes on, so that in course of time the loop of wire passes
right through the ice and falls off, but leaves the block of
ice in one piece ; a few small air bubbles imprisoned by the
freezing water mark the plane along which the wire has cut.

This melting under pressure and subsequent freezing
together again is spoken of as " regelation." It is well
illustrated by the behaviour of snow. Thus in making a
snowball the snow is pressed together and released to make
it " bind " ; the formation of a hard mass under the heel of
the boot when walking on snow is also an example of this
effect. If the temperature is much below freezing-point the
snow will not bind ; in other words, the pressure is not enough
to bring the melting-point as low as the existing temperature.

When a skater gets on to an edge, the area of the skate
blade in contact with the ice is very small, and hence the
pressure per unit area is large and the ice under the blade
melts, enabling the blade to sink in a little ; when it has
passed the water freezes again, so that there is no groove
left behind, but only a few imprisoned bubbles of air showing
where the skate has been, unless the ice is thawing. If the
temperature is very much below freezing-point, the skate will
not bite ; in other words, before it can sink in appreciably,
the area in contact with ice is such that the weight of the
body does not supply enough pressure to lower the melting-
point to the required extent. It may be noted that the
slipperiness of ice is due to this melting under pressure ;
very cold ice is not slippery.

Unlike water, the majority of substances occupy a greater
volume in the liquid than in the solid state, and consequently
the melting-point is raised by pressure. The Earth itself
probably supplies an example of this. It is well known that
the temperature of the earth increases with the depth below

144 HEAT

the surface, and if this increase is maintained the temperature
at depths quite small in comparison with the radius of the
earth must be above the melting-point of any known sub-
stance. All the evidence goes to show that the earth is solid,1
and it is concluded that the enormous pressure of the layers
above must raise the melting-point sufficiently to keep the
interior solid even at these temperatures. If for any reason
the pressure is released anywhere, liquefaction may take
place and volcanic phenomena occur.

The Freezing-point of Solutions. — The freezing-point
of a solution is lower than that of the pure solvent. For
example, the freezing-point of a solution of salt in water is
lower than 0° C. ; the exact temperature depends on the
strength of the solution. The freezing-point of a solution
of anything else in water is also lower than 0° C. ; different
substances produce different effects as regards the lowering.
It should be noted that when a solution freezes pure ice
separates out, leaving the solution more concentrated ; if the
freezing is continued, the solution may become so concentrated
that it reaches the saturation point when the dissolved
substance begins to separate also.

The Action of Ice and Salt — Freezing Mixtures. — We
have several times made use of a mixture of ice and salt in
order to obtain temperatures lower than 0° C. ; we are now
in a position to get some idea of what happens. This is a
particular case of a general principle depending on latent
heat.

In order co simplify the matter, let us consider what would
happen if we add ice to a solution of salt. Suppose we had
a solution containing about 14 per cent, of salt ; the freezing-
point of such a solution is -- 10° C. If the solution is at
— 10° C. and ice at — 10° C. is added, nothing much can
happen ; we have a solid in contact with its liquid at the
freezing-point, and there is no reason why any change should

1 For example, if the interior were liquid something equivalent to
tides would be produced ; but the occurrence of earthquakes regularly
twice a day is not a universal experience.

SOLID TO LIQUID 145

take place. But suppose they were both at 0° C. ; the solu-
tion is now in contact with ice at a temperature above its
freezing-point ; consequently some ice would melt. But in
order to do this it must get its latent heat of fusion from
somewhere, and it does it by taking heat from the solution
and itself and thereby lowers the temperature, which falls
below 0° C. This would go on until all the ice had melted
or the temperature had fallen to the freezing-point.1

Thus the solution in contact with ice has caused a lowering
of temperature because it makes the ice change its state
from solid to liquid.

This effect is quite a general one and is not confined to ice.
When any solid dissolves in water (or other solvent) it is
passing from the solid to the liquid and therefore there should
be a fall in temperature ; good examples of this are furnished
by ammonium nitrate and sodium thio-sulphate (" hypo "
of the photographer) which when dissolved in water cause
a considerable fall in temperature. When solid carbon
dioxide is dissolved in ether, a temperature below — 100° C.
can be obtained. It does not follow that a fall of temperature
will result in all cases ; there is also the action between the
solid and liquid to consider, and it may happen that the
heat evolved in this is greater than that absorbed to furnish
the latent heat of fusion, so that on the balance there is
a rise in temperature — potassium hydroxide dissolved in
water is a good example.

WTien salt is added to ice a certain amount of solution
occurs, and then the ice melts as already explained ; both
the solution of the salt and the melting of the ice involve
absorption of heat. Thus a fall of temperature results.
Absorption of heat can only go on as long as a change of
state is proceeding. The temperature cannot fall below about
— 20° C., because a solution of such strength that its freezing-
point is as low as this is also saturated with salt at that
temperature, so that it is impossible to get a salt solution

1 Since the melting ice would make the solution more dilute, the
freezing-point would now be above —10° C.

146 HEAT

with a freezing-point lower than this. The important thing
to remember is, that the effect of the salt is to cause the ice
to melt. For this reason it is a precisely similar operation
when salt is spread on a frozen pavement in order to get the
ice to melt : the ice does melt, but the temperature is
lowered.

Ice and salt form quite a good freezing mixture, but ice
and calcium chloride do better, since calcium chloride is
excessively soluble.

CHAPTER X
CHANGE OF STATE— LIQUID TO VAPOUR

If we take some water in a flask and heat it over a burner,
the temperature rises at first ; ultimately a stage is reached
when the water boils and no further rise in temperature takes
place. All that happens is that the water is converted to a
colourless vapour.1 Here we have another change of state,
and in some respects the phenomena are similar to those
accompanying the change from solid to liquid.

In the first place there is a very large volume change ;
1 c.c. of water forms about
1,700 c.c. of steam under
atmospheric pressure. The
engineer's rough rule is
that a cubic inch of water
forms a cubic foot of steam.
Secondly, a large amount of
heat is absorbed in turning the
water to steam. As a rough
illustration the following
experiment may be quoted : —

A flat -bottomed flask is
weighed empty, and then partly
filled with water and weighed
again. The flask is fitted with a cork through which pass a

FIG. 64.

1 When the vapour comes in contact with the cold air it condenses
to a visible cloud which is really a collection of very small particles of
water. This cloud is sometimes incorrectly spoken of as steam. Steam
itself is invisible as long as it is hot, e.g. in the upper part of the flask ;
when it becomes visible it is no longer steam but water.

147

148

HEAT

thermometer and a short outlet tube (Fig. 64). A fe\v
pieces of broken brick are put in to assist steady boiling
(see page 160) ; these are weighed with the empty flask.
The flask is placed on a gauze on a ring burner (a bunsen
would not supply heat at a rate sufficient for satisfactory
results) and the reading of the thermometer noted every
minute. When the water boils the heating is continued
for some time, and then the flask is removed and placed in
a dish of cold water. When it is sufficiently cooled the
outside is dried, the cork removed, and the flask and
contents weighed. The following figures illustrate what
happens : —

Weight of empty flask ..... 85 gms.

„ flask + water 470 gms.

Final weight of flask -f- water . . . .410 gms.

Original weight of water = 385 gms.

Weight of water boiled away = 60 gms.

Time.

Temp.

m'ns.

0

20° C.

1

28°

2

)

3

r

4

62° C.

5

73°

6

84°

7

95°

ill

100° C.

Surface of mercury hidden by the
cork.

Water boils. It is allowed to go on

boiling until
flask removed.

Now, in three minutes the temperature went from 28° to
62° C., or about 11^° per minute. The rate of rise afterwards
was 11° per minute. We may assume that the average rate
(neglecting the first minute) was about 11° per minute.

Since there were 385 gms. of water the burner was supplying
about 11 X 385 = 4,235 calories per minute to the water.
After the water boiled, heat was supplied for a further seven

LIQUID TO VAPOUR

149

minutes, and the result was that 60 gms. of water boiled
away.

Thus 7 X 4,235 calories were required to turn 60 gms. of
water at 100° to steam.

7x4 235

Thus the water required — p - calories per gm. to turn

it to steam,

= 494 calories per gm., or roughly 500.

Just as in the case of fusion, this heat is spoken of as latent.

Other liquids behave in the same way, but of course require
different amounts of heat.

Definition of the Latent Heat of Vaporization of a
Liquid. — The latent heat of vaporization of a liquid is the
amount of heat required to turn unit mass of the liquid to
vapour without change of
temperature.

Determination of the
Latent Heat of Vaporiza-
tion of Water or the Latent
Heat of Steam. — The method
described above must not be
regarded as in any sense
accurate. To obtain a reliable
value for the latent heat a
method somewhat similar to
that employed in the case of
ice is used. The idea is tq
pass steam into a known
weight of water in a calori-
meter and measure the amount
of heat given out by the
steam in condensing. The
calorimeter, of known weight
and containing a known weight
of water, is mounted on three
corks as before. In the mean-
time a boiler of the form shown Fio. 65.

150 HEAT

in Fig. 65 is got ready. It is made up of a copper vessel
with a narrow neck fitted with a cork ; through the cork
passes a glass tube which reaches nearly to the end of the
vessel. The vessel is half filled with water and placed on a
ring burner as shown. The burner stands on a sheet of
asbestos (with a small hole for the tube) carried on the ring
of a retort stand. The temperature of the water in the
calorimeter is noted, and then when steam is issuing freely
from the boiler the latter is transferred bodily by means of
the stand and placed with the end of the tube under the
surface of the water in the calorimeter. When the tempera-
ture of this water has been raised about 15 or 20° (but not
more x) the boiler is removed, the water is thoroughly stirred
and the highest temperature it attains is noted. The calori-
meter and its contents are now weighed,2 to find out the
weight of steam which has been condensed. The following
figures illustrate the method of calculation : —

Weight of calorimeter + thermometer . . 210 gms.

+ water . 778 gms.

Weight of water .... 568 gms.

Weight of calorimeter -f- thermometer -f- water -j-

condensed steam ...... 796 gms.

Weight of steam condensed . . 18 gms.

Initial temperature of water . . . 17° C.

Highest temperature reached . . . 35-5° C.
Water equivalent of calorimeter . . .19 gms.

Heat received by calorimeter and water —
= (568 -h 19) (35-5 — 17)= 587 x 18-5 = 10,860 calories.

If L is the latent heat of steam, the heat given out by the
steam in condensing at 100° C. is 18L calories.

1 If the temperature gets much above 35° C. not only is the cooling
correction serious, but considerable evaporation takes place, making
the apparent weight of the steam too small. The writer haa made
many experiments to verify the point.

2 It is best to keep the thermometer in for all the weighings ; if it is
taken out it may easily remove nearly a gramme of water, and since
the total weight of steam is not large, this forms a serious error.

LIQUID TO VAPOUR 151

Heat given out by the 18 gms. of water formed in cooling
to 35 5° is—

18(100 - 35-5) = 18 x 64-5 = 1,160 calories
18L + 1,160 = 10,860
18L= 9,700

L = 539 calories per gramme.

Tne object of having the boiler of the form shown is to
avoid as far as possible condensation taking place before the
steam gets to the calorimeter. If water is carried over with
the steam it is all weighed and counted as steam, and serious
errors result. An alternative method is to have some
form of steam trap on the tube, but the writer finds the
boiler as described to be more satisfactory ; it is easy to
move, and it is better to move the boiler than the calori-
meter.

The method given above cannot be relied on to more than
about 5 per cent. The weight of steam condensed is small
compared to the weight of water in the calorimeter, and
consequently this weight is determined by the difference
between the weights of two large quantities of water.1 More-
over, it is not. very satisfactory to have steam blowing over
the surface of the water as the nozzle is introduced and removed,
while a stoppage of the steam might result in some water
being drawn into the tube. For these and other reasons,
in accurate work the steam is condensed in a spiral tube of
metal contained in the calorimeter and is not allowed to mix
with the water ,• this method is also essential for finding the
latent heat of vaporization for liquids other than water.
Very elaborate and accurate experiments were performed by
Regnault ; a much simpler and quite accurate apparatus is
that due to Berthelot. The apparatus is shown in section
in Fig. 66. The flask has a wide tube down the centre which
is connected to the metal spiral and reservoir. It is heated

1 One may be permitted to describe an operation of this sort as
" weighing a haystack, putting a needle in, and weighing again in order
to find the weight of the needle."

152

HEAT

by a gas burner in the form of a circle with a gap in it, sc
that it can be withdrawn. The calorimeter is protected by
a thin wooden screen covered with a sheet of gauze. The
vapour passes down the wide tube into the spiral and con
denses in the reservoir. Since heat is conducted down this

tube, observations are made on
the rate of rise of temperature
of the water in the calorimeter
before the vapour comes over.
The calorimeter contains some
800 or 900 gms. of water, and the
rise in temperature was usually
3° or 4°. The time occupied is
from two to four minutes. The
apparatus was primarily intended
for determination of the latent
heat of vaporization of liquids
other than water ; M. Berthelot
tested its accuracy by finding the
latent heat of steam with it and
as his results agreed closely with
those of Kegnault he regarded its
accuracy as established.1

The value obtained by Reg-
nault for the latent heat of
vaporization of water at 100° C.2
is just under 537 calories per
gm. Callendar gives 539-3 calo-
ries per gm.

Effect of Pressure on the Boiling-point of a Liquid.—

The temperature at which a liquid boils is determined by
the pressure to which it is exposed. It is not a question of
minute effects as in the case of the melting-point ; at a

Fio. 66.

1 Berthelot, Comptes Rendus, LXXXV, 646, 1876.

2 The latent heat of vaporization is greater at low temperatures than
at high ; it vanishes at the critical temperature (see page 264).

LIQUID TO VAPOUR

153

pressure of two atmospheres the boiling-point of water would
be about 121° C. instead of 100° C. The boiling-point of any
liquid is raised by increased pressure ; this will readily be
understood when the meaning of boiling is considered (page
159).

The effect is well illustrated in the case of water by using
a Marcet boiler or Papin's digester (Fig. 67). This is a stout
copper vessel to which a
strong lid is attached by
studs. The lid is fitted
with a pressure gauge, a
lever safety valve, an
outlet tube with a tap
and a thermometer
pocket. A thermometer
pocket is simply a metal
tube closed at the lower
end, and reaching well
down into the vessel ; the
thermometer is placed in
the pocket and so is not
exposed to the pressure
inside the vessel. The
vessel is partly filled with
water and the weight ou
the safety valve lever is
put in such a position
that the valve will "blow
off " at a suitable pres-
sure . The water is boiled
by means of a ring FIG. 67.

burner or large bunsen,

and when the safety valve is blowing off the pressure on the
gauge is noted and the temperature as given by the ther-
mometer. The weight of the safety valve is then moved
so as to blow off at a higher pressure and the water boiled
again. In this way a series of readings may be obtained.

154 HEAT

It will be noticed that as the pressure is increased the boiling-
point is raised ; at a pressure of 200 Ib. per square inch 1 the
boiling-point is about 193° C.

In the same way, under low pressures, the boiling-point of
water is below 100° C. This may be illustrated by boiling
water in a Wurtz flask with its side tube connected to a vacuum
pump (an ordinary water-filter pump does very well). It
will be found that it is possible to get the water to boil at
quite low temperatures. The low temperatures at which
water boils under reduced pressure is also illustrated by the
well-known experiment due to Franklin. Water is boiled
in a round-bottomed flask, and after the steam has been
issuing for some time the flask is corked up and the flame
withdrawn. The flask is then inverted and supported on the
ring of a retort stand. During the boiling, the issuing steam
drove out most of the air, so that the space above the water
is occupied by vapour only. If now a little cold water is
poured on the flask by means of a sponge, some vapour con-
denses and the pressure in the flask is reduced. The water
thereupon boils vigorously for a few minutes until the pressure
is restored. Pouring on more cold water results in renewed
boiling, and the process can be continued for quite a long
while.

Determination of the Heights of Mountains by Means
of a Boiling-point Apparatus. — The atmospheric pressure
at the summit of a mountain is less than that at sea-level
by an amount which depends on the height of the mountain.
Hence a measurement of the pressure at the summit enables
the height to be calculated (the pressure at sea-level or some
known elevation must be known, since the atmospheric
pressure varies from day to day). A mercury barometer is
the most accurate means of measuring the pressure, but a
determination can be made by finding the boiling-point of

1 Tliis is the actual or absolute pressure, not 200 Ib. per sq. inch
above atmospheric ; it would be about 185 Ib. per sq. inch above the
atmospheric.

LIQUID TO VAPOUR 166

water. The boiling-point of water at different pressures has
been determined, and tables have been drawn up ; thus if the
boiling-point is known, the pressure can be ascertained. This
is the origin of the term hypsometer (height measurer) for a
boiling-point apparatus. A hypsometer is not so accurate
as a mercury barometer, but it is much more portable ; in
any case it serves as a useful check on an aneroid, and is
probably more reliable if the determination is carried out
carefully.

Evaporation. — Liquids can pass into vapour without
boiling. For example, if a shallow dish of water be left for
some time in a room the water will ul-
timately disappear ; another well-known
example is furnished when wet clothes
are put out in the open to dry. In this
case evaporation takes place, and in
order to get a clear idea of what happens
it will be convenient to put down the
known facts first, and then consider the
experimental work which established
them. Our knowledge of the facts is
founded on the work of Dalton. v

Let Fig. 68 represent a small dish of FIO. eg.

water underneath a large bell jar. Let
us suppose that the space under the bell jar is com-
pletely exhausted of air. - Then some of the water will turn into
vapour, and this will go on until the pressure of the vapour
inside the bell jar attains a certain value depending on the
temperature (this assumes that there is enough water for
this stage to be reached without all the water being used
up). For example, if the temperature were 15° C., evapora-
tion would go on until the pressure was 12-8 mm. of mercury.
The space is then said to be saturated with vapour, and
12-8 mm. is the saturation pressure of water vapour at 15° C.
If the bell jar were full of air or any other gas (so long as it
had no chemical action with water) the evaporation would

156

HEAT

still go on until the pressure of vapour was 12-8 mm. ; i.e.
if the pressure of air were originally 760 mm. the total
pressure would be 772-8 mm.1 The only difference would
be that it would take much longer to reach the steady
stage but the final result would be the same. It may
be noted that evaporation takes place
only at the surface (see page 276). The
saturation pressure increases with tempera-
ture; thus, at 20° C. it is 17-5 mm. for
water.

If, when the steady state has been
reached, the volume of the space were
altered, evaporation or condensation would
take place until the pressure of vapour
got back to its saturation pressure. If,
however, the volume were increased very
much a stage would be reached when
all the water had evaporated ; further
expansion would result in the space being
unsaturated. The passage from liquid to
vapour involves the absorption of heat,
as we have already seen ; if this is not
supplied from the surroundings the liquid
will cool as it evaporates.

Measurement of Vapour Pressures. —

The most straightforward way of measuring
the vapour pressure of a liquid at moderate
FIG 69 temperatures is to make use of a barometer

tube. For example, to measure the vapour
pressure at room temperature, a barometer tube is filled
with mercury and inverted in a bowl of mercury. The
height of the mercury column (measured from the surface
of the mercury in the tube to the surface of the mercury

1 This result, known as Dalton's Law, that the pressure of the vapour
is independent of the substances already occupying the space, is found
not to be strictly true at high total pressures.

LIQUID TO VAPOUR 157

in the bowl) is noted. A little of the liquid is then intro-
duced by means of a pipette bent at the end as shown in
the section Fig. 69.

The liquid rises to the top of the column and evaporates
in the vacuous space ; enough liquid must be sent up to leave
some remaining when the evaporation is complete. The pres-
sure of the vapour drives the column down a certain amount ;
the height of the column is measured, and the difference
between this height and the original gives the pressure of the
vapour. If there is a large amount of liquid on the top of
the mercury, a correction must be made for the pressure
due to the weight of it ; if the length of the liquid column is
I and the specific gravity d, it will be equivalent to an extra

height of mercury of — — , taking 13'6 as the Specific Gravity
lo'b

of Mercury. Now, suppose the original height were 760 mm.
and the height afterwards was 748 mm., the pressure of the
vapour would be 12 mm. of mercury, since the atmosphere
is now supporting a column of mercury 748 mm. + the
pressure of the vapour. It can be seen that adding more
liquid does not alter the pressure of the vapour (correction
must be made for the extra column of liquid).

By having several barometer tubes the vapour pressure of
various liquids can be measured. Thus at 15° C. the vapour
pressure of water is 12-8 mm. of mercury ; the vapour pres-
sure of alcohol is 32-6 mm. of mercury ; the vapour pressure
of ether is about 357 mm. of mercury.

Vapour Pressures at Different Temperatures. — The
vapour pressure of a liquid increases as the temperature is
raised ; this is well illustrated in the case of water by means
of the apparatus described below, which can also be used for
other liquids. Fig. 70a shows the arrangement for tempera-
tures below 100° C.

B is a straight barometer tube which is connected to a
tube C of the same bore but open at the top. The arrange-
ment forms a syphon barometer, and into the Torricellian
vacuum some water has been introduced. The two tubes

158

HEAT

To

JBoiler

are surrounded by a wide tube, which is filled with water, the

outlet D being closed with a clip.

The water in the jacket tube can be raised to any
convenient temperature by blow-
ing in steam, the water is then
thoroughly stirred, the tempera-
ture noted, and the height h
measured.

The pressure of the vapour is
then the pressure of the atmosphere
minus pressure due to a column
of mercury of height h. For
rough work we can get the vapour
pressure by simply subtracting h
from the reading of the barometer
in the room ; for more accurate
work the height h must be cor-
rected for the density of the mer-
cury at the temperature of the
jacket.

By blowing in steam the water
can be raised to a higher tempera-
ture, and in this way a series of
readings can be obtained. It will
be observed that the increase in
vapour pressure is not proportional
to the rise in temperature, but is
more rapid at high tempera-
tures. •

The jacket is then emptied by
opening D ; a cork is fitted at the
top as shown in Fig. 706, and steam
is passed through. It will then be
observed that the mercury in the
two tubes comes to the same level

(correcting for the weight of any water on the top of

the mercury in the closed limb), so that at the boiling-point

LIQUID TO VAPOUR 159

the pressure of the water vapour is equal to the atmospheric
pressure.

This result is most important, and will be found to be the
case for all liquids ; for example, if we had alcohol in the
tube B and passed vapour from boiling alcohol through the
jacket, we should get the same effect. At temperatures above
the boiling-point the vapour pressure would be above the
atmospheric, so that the mercury in C would stand higher
than that in B. We are now in a position to understand the
meaning of boiling.

Ebullition or Boiling. — Suppose we have a beaker of
water being heated by a burner. As the heat is supplied the
temperature of the water rises. A certain amount of evapora-
tion takes place at the surface, and faint clouds appear a
little above the surface where the warm vapour has cooled in
contact with air. This evaporation involves the supply of
latent heat of vaporization, but the burner supplies heat at
a rate which more than keeps pace with this (if it does not, a
stage is reached where the vapour carries off all the available
supply of heat and no further rise of temperature takes place)
and the temperature continues to rise. During this stage
some of the air or other gases dissolved in the water begins
to come out and form small bubbles, which usually cling to
the side of the vessel. But when the temperature is such
that the vapour pressure is equal to the atmospheric, a new
possibility comes in ; for bubbles of vapour can form within
the liquid l and grow by evaporation. As these bubbles
provide evaporating surfaces (from the liquid into the interior
of the bubble) enough evaporation takes place to use up
the whole heat supply ; more rapid supply of heat would
merely give more bubbles. At this stage the liquid is said
to boil, and the characteristic of boiling is the formation of
vapour at points within the liquid as well as at the free
surface.

1 They require something to start on, which is furnished by some of
the gas bubbles or a roughness on the side of the vessel ; they do not
start in the midst of the liquid, but from some object.

160 HEAT

Since the evaporation into the interior of the bubble has
bo overcome the pressure of the liquid on the outside, it is
clear that the bubble cannot grow unless the vapour pressure
is at least equal to this outside pressure. Hence the liquid
cannot boil until the vapour pressure is equal to the atmo-
spheric pressure. It therefore becomes obvious why the
boiling-point depends on the external pressure.

Delayed Boiling. — If there are no suitable objects for the
bubbles to start on, it is possible for the temperature of the
liquid to be raised considerably above its proper boiling
point ; if a bubble is then formed it grows violently, and
the effect known as " bumping " occurs. To prevent this,
small pieces of a porous substance, such as pieces of broken
brick, should be put into the liquid to form suitable nuclei
for the bubbles (compare page 148).

The Boiling-point of a Solution. — The presence of a
substance l dissolved in a liquid lowers the vapour pressure,
so that if the vapour pressure is to attain the pressure of the
atmosphere the solution has to be raised to a higher tempera-
ture than would be required for the pure liquid. In other
words, the boiling-point of a solution is higher than that of
the pure liquid. (It is this lowering of the vapour pressure
which lowers the freezing-point, but the connection is not so
obvious at first sight.)

Determination of Boiling-points. — The procedure de-
pends on whether we are dealing with a pure liquid or
a solution. The following experiments will make this
clear.

A flask is taken, fitted with a cork through which pass a
thermometer and an outlet tube. Some pure water is placed
in the flask, together with a few pieces of broken brick. The
thermometer is arranged to stand with its bulb an inch or

1 We are here considering the case of substances which are not
appreciably volatile, so that, the vapour given off is that of the solvent
only. With a solution of a volatile substance each lowers the vapour
pressure of the other so that the resulting vapour pressure is lower
than the sum of the vapour pressures of the two -separate substances.

LIQUID TO VAPOUR

161

so above the surface of the water. The water is boiled, and
when the steam is issuing freely and the reading of the thermo-
meter is steady the temperature is noted. The experiment is
repeated with bulb of the thermometer in the water ; the
temperature will be found to be much the same as before.
It may be a trifle higher owing to delayed boiling or traces
of dissolved substance.1 These experiments are repeated
using a solution of salt in water. It will be found that the
temperature of the vapour is the same as it was for pure
water,2 while the temperature of the liquid may be several
degrees higher.

Thus if we are determining the boiling-point of a pure
liquid, the thermometer should be in the vapour ; this will
avoid any possibility of an error due to delayed boiling or
other causes. An apparatus of the hypsometer type should
be employed.

If we are determining the boiling-point of a solution the
thermometer must be in the liquid, otherwise we shall get
the boiling-point of the solvent, not of the solution.

Determination of the Boiling-point of a Liquid when
only a small quantity is available. — A
U-tube is taken, having one limb sealed
and the other open (Fig. 71). It is par-
tially filled with mercury as shown, and a
few drops of the liquid introduced as shown
at A. The tube is placed in a bath of some
liquid of higher boiling-point, and the tem-
perature gradually raised. A stage is reached
when the mercury stands at the same level
in each limb ; the temperature at which
this takes place is the boiling-point of the
liquid A at the pressure of the atmosphere
at the time of the experiment.

FIG. 71.

1 There is also a smaller length of the thermometer stem exposed.

* If the first reading is taken with the thermometer in the solution
eome salt will remain on the bulb and make the vapour appear to have
a higher temperature unless the bulb is carefully washed.

M

162

HEAT

Vapour Pressure of Liquids at High Temperatures.
Regnault's Method. — The pressure in the reservoir B (Fig.
72) was maintained at some definite pressure, as indicated
by the manometer C, by pumping in or with-
drawing air. The liquid was heated at A and
ultimately boiled, the vapour condensing at
D and flowing back to A. The thermometers *
gave the temperature at which the liquid
boiled under the pressure in the reservoir, i.e.
the temperature at which the vapour pres-
sure was equal to that registered by C. The
pressure in B was then increased, and the
reading of the thermometers gave the boiling-
point under the new pressure. In this way
Regnault determined the vapour pressure of

B

FIG. 72.

FIG. 73.

water at temperatures ranging from 50° C. to 230° C., at
which latter temperature the pressure was 27 atmospheres.

Vapour Pressure at Low Temperatures — Evapora-
tion from Solids. — Ice and many other solids have appreci-

1 The thermometers gave the same reading, indicating an absence of
delayed boiling.

LIQUID TO VAPOUR 163

able vapour pressures. Regnault determined the vapour
pressure of ice by means of the apparatus shown in Fig. 73.
The barometer tube is bent over and terminates in a bulb A
which contains the ice. A is immersed in a freezing mixture
which is liquid and well stirred. The solids have a definite
vapour pressure corresponding to each temperature. The
pressure of the atmosphere is read off on B.

Sublimation. — If a solid such as iodine is heated in a tube
it will vaporize without liquefying and the vapour will con-
dense back to solid in the upper part of the tube. Such a
process corresponds to the distillation of a liquid and is known
as sublimation.

The Properties of an Unsaturated Vapour. — The
general behaviour of a vapour may be illus-
trated in the following way. Suppose we have
a long barometer tube in a tall trough of mer-
cury. Let a very small quantity of water or some
other liquid, e.g. alcohol or ether, be introduced.
Starting with the barometer tube at the bottom of
the trough, let it be gradually raised. At first
the mercury, with the liquid on the top, fills the
tube ; a stage is soon reached, as shown in Fig. •*••
74, at which a space develops into which the
liquid has evaporated. Here the height h is
such that the difference between the barometer
reading and h is equal to the vapour pressure of
the liquid. As the tube is raised the mercury
remains at the same height h ; the volume of
the space is increased, but more liquid evaporates
and the vapour pressure remains constant, as long
as any liquid is left. But when all the liquid FIG. 74.
has gone, further raising of the tube results
in the height h of the mercury increasing. The space is
now unsaturated, and the unsaturated vapour behaves like
any gas ; as the volume is increased the pressure is diminished.
It does not, however, obey Boyle's Law very accurately, but
as the pressure diminishes the vapour approaches more nearly

164 HEAT

bo obeying the Law. If the tube is lowered, the reverse
process goes on ; the pressure increases as the volume
diminishes, until a stage is reached when the saturation
pressure is reached and liquid begins to form.

If we take a gas like sulphur dioxide and compress it, a
stage is reached at which the relation between pressure and
volume deviates considerably from Boyle's Law, and if we go
on compressing it the gas will ultimately liquefy. The process
is precisely similar to that of the preceding paragraph, but
the pressures involved are much greater ; for example, at
15° C. sulphur dioxide requires a pressure of about 2-7 atmo-
spheres to liquefy it and carbon dioxide about 55 atmospheres.

It is found, however, that at ordinary temperatures no
amount of pressure will liquefy gases like hydrogen, oxygen,
or nitrogen. Pressures up to 30,000 atmospheres have been
used. Only when they are cooled to very low temperatures
indeed can they be liquefied. There is, however, no funda-
mental difference between these gases and others, for it is
found that for any gas there is a certain temperature, called
the critical temperature, above which no amount of pressure
will liquefy it. In the case of hydrogen and oxygen the
critical temperature is very low.

Thus the critical temperature for hydrogen is — 234° C

,, ,, ,, ,, sulphur dioxide is 155° C

,, ,, „ ,, carbon dioxide is 31-1° C

,, „ „ ,, water is about 365° C

The question of the liquefaction of gases is dealt with in
Chapters XVII and XIX.

Cooling due to Evaporation. — Since the change from
liquid to vapour involves an absorption of heat, it follows that
if a liquid evaporates and heat is not supplied, the liquid must
take heat from itself and the surroundings. This may readily
be illustrated in the case of a volatile liquid like ether. Some
ether is placed in a flat- bottomed flask and the flask is placed
on a block of wood ; a few drops of water are put on the block

LIQUID TO VAPOUR

165

before the flask is put on. A stream of air is blown through
the ether by means of foot bellows ; this causes rapid evapora-
tion, and in a few minutes the water under the flask will be
frozen, so that the block can be lifted by the flask.

Water can be made to freeze by its own evaporation ; to
do this the evaporation must be rapid. Rapid evaporation
can be brought about by having a dish of water in an ex-
hausted enclosure and absorbing the vapour as fast as it
comes off, so that the space does not become saturated. A
few drops of water are placed in a watch-
glass supported above a porcelain vessel con-
taining concentrated sulphuric acid ; sulphuric
acid has a strong affinity for water. The
whole is placed under the receiver of a good
air pump. The pump is worked and the water
evaporates ; after a time the remaining water
will freeze into a solid lump.1

Wollaston's cryophorus is an illustration of
the same effect. This instrument is a glass tube
with a bulb at each end (Fig. 75). It contains
some water, and before sealing off it is ex-
hausted, so that the space not occupied by
water contains water vapour only. One bulb,
B, is placed in a freezing mixture of ice and
salt. This condenses the vapour in B and
lowers the pressure of vapour in the tube.
Water in A evaporates to restore the pressure ;
more vapour is condensed at B and more water
evaporates in A. This process goes on, and
the evaporation lowers the temperature of the
water in A until it finally freezes. It is advisable to wrap
the bulb A in flannel or other poor conductor, otherwise heat
is absorbed from the air and the water takes a very long
time to freeze, and may even fail to do so. It will be ob-
served that the process is precisely similar to that described

1 If the under side of the watch-giaas is wet, the experiment seldom
succeeds.

B

FIG. 75.

166

HEAT

in the previous paragraph, except that the vapour is removed
by condensation instead of by sulphuric acid.

The danger of standing about in wet clothes is due to the
great absorption of heat on evaporation ; anything that
increases the evaporation, such as a wind which carries away
the vapour as it forms, increases the danger. Compare the
advice to a benighted party given in the Badminton volume
on Mountaineering : " The cold due to evaporation is much
more to be dreaded than a low temperature. If any man
then has anything that is dry, let him put it on. A Scotch
shepherd, if his plaid is partly wet through and he is caught
at night, turns his plaid with the wet side in." 1

Refrigerating Machines. — The cooling due to rapid

FIG. 76.

evaporation is made use of in many refrigerating machines.
The liquid is made to evaporate rapidly in a spiral metal pipe
surrounded by a vessel containing strong brine, which can be
cooled below 0° C. without freezing. The substances to be
cooled are placed in the brine, e.g. vessels containing water
can be put in, and the water will be frozen to a solid block
of ice.

It is usual to employ either ammonia gas or carbon dioxide
liquefied under pressure as the evaporating liquid, and the
process is rendered continuous by an arrangement illustrated
diagrammatically in Fig. 76.

1 The present writer is unacquainted with the habits of Scotch shep-
herds, but the principle is sound, and he has often applied it.

LIQUID TO VAPOUR

167

The gas is compressed by the pump P and liquefies in the
pipe A, which is kept cool by circulating water. The liquid
is admitted through the regulating valve V into the spiral B.
It evaporates there, the pump drawing off the vapour and
re-delivering it to A. The process is thus continuous,
evaporation constantly going on in B, which forms the
refrigerating part.

For most purposes ammonia is the most convenient sub-
stance. Carbon dioxide involves much higher pressures ; it
has the advantage, however, that the apparatus can be
smaller, and if the gas should escape by an accidental burst,
carbon dioxide is less injurious than ammonia. Refrigerating
machines are examples of reversed heat engines or "heat
pumps" (see Chapter XXI).

Change of State as a Means of Measuring Quantities
of Heat.

Instead of measuring a quantity of heat by finding the
change in temperature of a known mass of water, we can
make the heat melt ice or vaporize water, and if the amount
of ice melted or water vaporized be measured, and the latent
heat is known, the amount of heat is readily calculated.

Ice Calorimeters. — The most satisfactory of these
is Bunsen's (Fig. 77). The
amount of ice melted is
determined indirectly, by
making use of the change Q
in volume when ice turns
to water. The apparatus
consists of a glass bulb Q,
into the top of which is
fused a tube P, and which
terminates in a tube. This
tube is provided with a
collar S into which a narrow
tube R fits. The upper

*x

part of Q contains water,

FIG. 77.

168

HEAT

carefully freed from air, while the lower part and the tube S
contain mercury. Some of the water in Q is frozen by draw-
ing alcohol, cooled in a freezing mixture, through P. In
this way a cap of ice is formed round P. Some distilled
water is placed in P, and then the whole apparatus is sur-
rounded by melting ice or snow and left for a long time so that it
takes up a temperature of 0° C. The substance whose specific
heat is to be measured is heated to a known temperature, e.g.
in a steam heater, and dropped into P. It cools down to
0° C. and melts some of the ice. The contraction in volume
causes the mercury in R to move. The distance moved is
measured and the contraction in volume is thus found if the
bore of the tube R is known. From this the amount of ice
melted is known, and so the amount of heat given up by the
solid in cooling to 0° C. is found. In practice the apparatus

is usually calibrated directly
by measuring the movement
of the mercury in C when a
known mass of hot water at
a known temperature is
placed in D.

It will be observed that
there is no correction for
the water equivalent of the
calorimeter since this starts
and ends at the same tem-
perature ; nor is there any
cooling correction. The ap-
paratus is by no means easy
to use, but it gives good
results and is well adapted
for finding the specific
heats of substances when
only a small quantity is
available.

Joly's Steam Calori-
FIG. 78, meter. — In this apparatus

LIQUID TO VAPOUR 169

(Fig. 78) the idea is to place the substance in a pan suspended
from one arm of a balance and enclosed in a chamber into
which steam can be passed.

The substance is placed in the pan and left for some time
so as to take up the temperature of the chamber, which is
given by a thermometer. Steam is then passed through the
apparatus as shown by the arrow. Some of this steam con-
denses on the substance, raising its temperature ; as soon as
its temperature reaches the boiling-point no more steam
condenses and the weight becomes constant. The increase
in weight gives the weight of steam condensed.1 The sub-
stance rests on a network of threads so as not to touch the
pan. Otherwise so much steam will condense on the lower
side of the pan that a little water may drop and be lost.
An alternative is to have a catch-water attached to the
underside of the pan.

Now, if w grams of steam were condensed, 537 X w calories
have been used in raising the temperature of the substance
and pan from the initial temperature of the chamber to the
boiling-point of water. A preliminary experiment is made to
find out the thermal capacity of the pan and supports, the
weight of steam condensed being found when there is no
substance in the pan.

A spiral of platinum wire made red hot by an electric
current prevents steam condensing and forming a drop of
water where the supporting wire passes out, which would
interfere with the weighing.

The method is quick and very accurate ; it is very suitable
for dealing with small quantities of material. Solids or liquids
which would be affected by steam can be sealed up in thin
glass vessels ; a correction for the thermal capacity of this
vessel must be made.

Specific Heats at Low Temperatures. — The substance

1 A correction is necessary, because there is an apparent change in
weight owing to the first weight of the substance being taken in air,
and the second in steam, which is less dense than air and consequently
exerts a smaller upthrust on the body.

170 HEAT

is first brought to a definite temperature, say that of liquid
nitrogen at atmospheric pressure, and then dropped into
liquid hydrogen, which is at a temperature much lower than
that of liquid nitrogen. The amount of hydrogen evaporated
is found by measuring the volume of the gas given off, and
hence the heat given up to it is calculated. The liquid
hydrogen vessel is surrounded by a jacket filled with liquid
hydrogen also to prevent the evaporation of hydrogen in
the vessel due to absorption of heat from its surround-
ings.

CHAPTER XI
HYGROMETRY

Evaporation is constantly taking place from the surface
of the sea and inland water, and a notable amount takes
place from the surface of the leaves of trees and plants ; on
the other hand, water returns to earth in the form of rain,
dew, etc. Thus the air always contains more or less water
vapour as one of its constituents ; the amount varies according
to circumstances. A knowledge of the amount of water
vapour in the air is of importance in meteorology, and the
measurement of it is termed hygrometry.

The presence of this vapour is easily demonstrated. If a
glass flask or a metal vessel is filled with water cooled by a
little ice, a film of moisture is soon formed on the outside.
A piece of fused calcium chloride exposed to the air absorbs
moisture and becomes pasty and finally liquid, forming a
solution of calcium chloride in water. The same effect may
be shown in another way by drawing air through a U-tube
containing fused calcium chloride or glass beads moistened
with strong sulphuric acid. If the tube be weighed before
and after, it will be found to have gained in weight owing
to absorbed moisture.

We saw in Chapter X that at any particular temperature
a given space can only contain a certain amount of water
vapour whether air is present or not ; if there were more
vapour the pressure would be above the saturation pressure
and some vapour would condense.1 We also saw that the
amount of water vapour required to saturate a given volume

1 But see page 183 on the necessity for some nucleus on which to
condense.

171

172 HEAT

is much greater at high temperatures than at low ; thus warm
air can contain a greater quantity of aqueous vapour per unit
volume than can cold air. As a general rule the amount of
aqueous vapour in the air is less than that required to saturate
it, and the Relative Humidity of the air is denned as —
Amount of aqueous vapour per unit volume

Amount of aqueous vapour per unit volume required to

saturate the air at the same temperature.
Formation of Clouds and Mist. — If a mass of air, which
contains less vapour than is required to saturate it, is quickly
cooled, the amount of vapour which it contains may exceed
that required to saturate it at the lower temperature, and
condensation takes place throughout the mass. A large
number of small drops are formed which constitute a cloud
or mist. There are two very common ways in which cooling
may take place : —

(1) A current of warm air carrying a good deal of aqueous
vapour may meet a colder current, and condensation takes
place where they meet. The fog-banks off the coast of
Newfoundland are usually quoted as examples of this.

(2) A much more general effect is the case of a mass of
air passing rapidly up into a region of lower pressure, e.g. a
wind blowing against mountain sides and deflected upwards.
The mass of air expands, and when a gas expands it does
work (see Chapter XIX) involving absorption of heat. The
cooling may be sufficient to cause condensation.

Formation of Dew. — The formation of dew is a case of
local condensation. If we have a cold object, the air in con-
tact with it will be cooled, and may reach a temperature at
which the vapour in it is sufficient to saturate it, and it will
begin to deposit moisture on the object. The effect will take
place much more readily if the object is on or near the ground ;
if it is not, the air as it cools, becoming denser, will sink before
it has reached the saturation-point and more air will take its
place. In the same way the effect will be hindered if the air
is not still, since a given portion of air will not remain in
contact with the obiect.

HYGROMETRY 173

Now, on a clear night radiation takes place freely from the
surface of the earth, so that objects on the earth become
very cold.

If the sky is cloudy, the clouds act as a screen, sending
back a good deal of the radiation. Thus dew forms more
readily on a clear night than on a cloudy one. The best
conditions for the formation of dew are provided when a hot
day is followed by a clear still night. During the day
evaporation into the atmosphere takes place readily, since
the air can contain a great deal of aqueous vapour before
reaching its saturation-point ; while the clear night favours
radiation.

It may be noted that the heavy deposit of dew on grass
and the leaves of plants is not wholly produced, by deposition
from the atmosphere ; we have seen that the leaves give off
water, and if the air in contact with them is already cooled
to the saturation-point this water cannot pass off as vapour,
but appears as liquid.

Hoar Frost. — If the objects are at a temperature below
freezing-point any condensation will be direct from vapour
to solid. Thus, hoar frost is not frozen dew, but ice formed
directly from vapour.

Condition of the Air . Relative Humidity . — In meteoro
logical work a knowledge of the condition of the air is of value ;
if the air is nearly saturated very slight cooling will bring
about condensation. It will be seen that what we want to
know is not so much the actual amount of vapour in the air
as the relative humidity. This is also a guide as to the
general feeling of the air ; if the relative humidity is low,
evaporation readily takes place into the air, which in conse-
quence feels " dry," while if the relative humidity is high —
i.e. the air is nearly saturated — very little evaporation can
take place and the air feels damp. It may often happen
that the air on a hot summer's day contains much more vapour
than on a cold damp day in winter, but since the amount
required to saturate the hot air is so great, the relative
humidity is lower and the air feels dry.

174 HEAT

Measurement of Relative Humidity. — We have defined
relative humidity as —

Amount of water vapour per unit volume

Amount of water vapour per unit volume required to saturate

the air at the same temperature.

Instead of expressing the amount of water vapour as so
many grams per unit volume it is often more convenient to
express it in terms of the partial pressure it exerts.

Properties of a Mixture of Gases and Vapours. —
Partial Pressures. — The experiments of Dalton, confirmed
by the much more elaborate work of Regnault, showed that
if we have a space occupied by a mixture of gases and vapours,
which do not have any chemical action on one another, the
total pressure is equal to the sum of the pressure which each
constituent would exert by itself if it occupied the whole
volume. For example, if we had 1 litre of nitrogen, measured
at 500 mm. of mercury, 1 litre of oxygen measured at 250 mm.,
and 1 litre of water vapour measured at 10 mm., and put
them all into a 1 -litre vessel, the pressure of the mixture
would be 760 mm.1

It is easy to see that this is so, if each constituent obeys
Boyle's Law. Thus we may regard each constituent as
diffused through the whole volume, and contributing to the
total pressure a partial pressure equal to that which it would
exert if the other constituents were not there. Regnault
also made experiments which showed that although water
vapour does not obey the laws of Boyle and Charles so closely
as a gas like nitrogen or oxygen, the departure is quite small
for moderate pressures except near saturation point. Even
here later work shows that the deviation at moderate tem-
peratures is not more than about 1 or 2 per cent. In

1 This is not strictly true, and experiments made at high pressures
show considerable deviations from this law. This is to be expected
when the molecules are closely packed (see Chapter XVIII). At pressures
of the sort met with in the atmosphere the law is sufficiently near the
truth for all practical purposes and is assumed to be true in meteoro-
logical calculations.

HYGROMETRY 175

Hygrometry, therefore, it is usually assumed that water
vapour obeys these laws up to the saturation point.
Relative humidity is therefore also denned as —
Pressure of aqueous vapour actually present

Saturation pressure of aqueous vapour at the same
temperature.

Determination of Relative Humidity by the Dew-
point Method. — If a portion of the air in an open space be
cooled, the cooled air contracts but the pressure remains
constant, equal to the atmospheric. The pressure of the
water vapour in it also remains constant, since the vapour
contracts in the same proportion as the air. A stage is
reached, however, at which the temperature is such that the
pressure of the water vapour corresponds to the saturation
pressure ; if the cooling is continued condensation will take
place and the pressure of the vapour will fall. This tempera-
ture, when condensation can just take place, is called the
dew-point, and it will be seen that the pressure in the air
before cooling is equal to the saturation pressure at the
temperature of the dew-point.

For example, if the dew-point were found to be 8° C., we
see from the tables that the saturation pressure at this
temperature is 8-6 mm. Then we can say that the pressure
of vapour in the air before cooling was 8-6 mm.

Daniell's Hygrometer. — This instrument, the oldest
form of hygrometer, is sometimes used, but is now recognized
as unreliable.

Dines' Hygrometer. — A very simple hygrometer is that
due to Dines (Fig. 79). A reservoir A contains water cooled
by ice. It communicates with a thin metal box B closed at the
top with a thin sheet of black glass C. The thermometer D
has its bulb just under the glass plate, and its stem in a groove
in the wooden block which supports the parts of the apparatus.
The box B is full of water at room temperature. By opening
the tap T ice-cold water from the reservoir is allowed to flow
into B which is provided with an outlet E. Thus the glass

176

HEAT

plate is gradually cooled, until a deposit of dew is seen ; the
temperature is noted, and the flow of cold water stopped.
After a time the dew disappears as the plate gradually be-
comes warmer, and the temperature at which this happens
is noted.

The mean of these two temperatures, if they do not differ
much, is taken as the dew-point. If they do differ much,
the experiment must be repeated and the cooling carried
out more slowly, so that the two temperatures do not differ
by more than one or two tenths of a degree.

It is important that the observer should not be too near,

FIG. 79.

otherwise his breath will affect the humidity of the air. The
thermometer can be read by a telescope and inclined mirror ;
observation of dew is facilitated by viewing the black glass
by means of a beam of light reflected from the surface.
Failing these precautions a large sheet of glass must be
interposed between the observer and the instrument.

Regnault's Hygrometer. — Regnault's hygrometer is very
satisfactory. The apparatus consists of a glass tube A (Fig.
80), to the lower end of which is cemented a capsule made
of thin silver, highly polished on the outside. The glass tube
is closed at the top by a cork through which passes a tube
reaching nearly to the bottom of the apparatus. A side tube
communicates through the tubular support with a long rubber
tube connected to an aspirator. Ether is placed in the
apparatus, and air is drawn through it by means of the
aspirator. By this means the silver capsule is cooled, and

HYGROMETRY

177

the temperature at which dew is deposited is noted. The
flow of air is stopped and the temperature at which dew
disappears is noted.

The thermometer is read by means of a telescope, and the
rate of cooling can be completely controlled by means of the
aspirator, which is some
distance off ; the observer
therefore need not come near
the apparatus. The bub-
bling of the air through the
ether acts as a stirrer, so
that the temperature of the
whole of the ether is given
by the thermometer.

It is not very easy to see
the deposit of dew, and ob-
servation is facilitated by
having a similar tube B for
comparison. Air is not
drawn through this tube.

The two forms of hygro-
meters described only give
good results if the air is
still ; if there is a draught
the formation of dew is
prevented unless the tem-
perature is much below the true dew-point.
v The Wet and Dry Bulb Hygrometer. — A form of hygro-
meter which is very much in use is the wet and dry bulb
hygrometer (known in this country as Mason's hygrometer).
Two thermometers, of the same shape and size, are mounted
side by side, but the bulb of one of them is enclos3d in a
piece of muslin which is kept moist by means of pieces of
cotton wick dipping into the small vessel of water J (Fig. 81).

1 Details of the most satisfactory arrangement are given in The
Observers' Handbook, published by the Meteorological Office, which
also contains Glaisher's Tables.

FIG. 80.

178

HEAT

As evaporation takes place from the muslin, heat is absorbed,
and the wet bulb thermometer indicates a lower temperature
than the other. The difference in temperature depends on
the rate of evaporation, which again depends on the relative
humidity of the atmosphere ; if the air is saturated, the two
thermometers give the same reading. Tables have been

drawn up, from which the
dew-point may be obtained
if the dry bulb reading and
the difference of the readings
of the wet and dry bulb are
known. Since the rate of
evaporation would be in-
creased by motion of the air,
the instrument is usually
placed in a Stevenson screen,
which is essentially a cham-
ber with double louvre sides,
i.e. sides arranged like a
double row of Venetian blinds.
The instrument is quickly
and easily read, but cannot
be regarded as accurate. The
readings are appreciably in-
fluenced by the surroundings ;
and it does not seem possible
to deduce a relation between
FIG. 81. the readings and the actual

pressure of water vapour in

the air. The only reasonable method appears to be to com-
pare the readings of the instrument with those of one of
the other types.

Modern Forms. — In order to avoid the uncertainties caused

by air currents, modern hygrometers of the wet and dry

bulb type are usually provided with an arrangement by which

the air is drawn over the bulbs at a known constant rate.

A type of dew-point instrument which is reliable has been

HYGROMETRY 179

designed by M. Crova. Instead of letting the dew be deposited
on the outside of a cooled vessel, Crova has a thin brass tube,
closed except for an inlet and outlet, which is gradually cooled
down from the outside while air is slowly drawn through it.
When the tube reaches a low enough temperature, dew is
deposited on the inside and is made visible by an ingenious
optical arrangement. For particulars of this instrument
reference may be made to Preston's Heat ; others are des-
cribed in makers' lists such as those of Negretti and Zambia.

Glaisher's tables, used by the Meteorological Office, are
based on a long series of comparisons with a Daniell hygro-
meter ; we have already seen that this is not a trustworthy
instrument. The wet and dry bulb hygrometer must there-
fore be regarded as an extremely convenient instrument for
giving a rough value of the humidity of the air at any time.1
Moreover, since it can be enclosed in a Stevenson screen, it
is suitable for use out of doors, under conditions where Dines'
and Regnault's could not be used. It is therefore the instru-
ment used at meteorological stations.

The Chemical Hygrometer.— The chemical hygrometer is
simply an arrangement in which a known volume of air is drawn
through drying tubes, and the weight of aqueous vapour is
determined directly. It is very accurate but somewhat tedious,
and it should be noted that it does not give the state of the air
at some particular moment, but the average condition during
the time of the experiment.

Fig. 82 shows the arrangement. A and B are drying tubes
containing pumice moistened with strong sulphuric acid ; these
are weighed before and after the experiment. 2 P is a small tube
containing phosphorus pentoxide • ; it is weighed before and
after and is used to see that A and B are really absorbing all the
water vapour. P should not change in weight ; if it does the

1 The compilation of more reliable tables for the wet and dry bulb
hygrometer is at present receiving attention.

2 During the weighing and at all times except while the air is being
drawn through them, the side tubes of the U- tubes are closed with glass
plugs.

* P is known as a " policeman," since it is there to see that A and B
perform their office.

180

HEAT

experiment must be done again. Air is drawn through by means
of the aspirator E, and the drying tube C is used to prevent any
aqueous vapour diffusing back from E into the weighed tubes.
The syphon tube is filled before connecting up the drying tubes,
and is then closed by the screw clip D. When the tubes are
connected D is partially opened and the air is slowly drawn
through ; several litres are allowed to pass, the volume of the air
which has come into the aspirator being determined by the
volume of water which flows out.1

If an accuracy of 1 or 2 per cent, is sufficient, the relative

FIG. 82.

humidity of the air can be obtained by a very simple calculation.
Let w be the weight of water vapour, and V litres the volume of
air as measured by the amount of water which flowed out of the
aspirator. If T«, is the temperature of the air in the aspirator,
determined by taking the temperature of the water, and TA the
temperature of the air outside we can assume that the volume of

O»7O 1 rp

the air which passed through the drying tubes is V . 970 , rp —

1 Care must be taken to start and finish at atmospheric pressure ;
at the finish the measuring cylinder is raised until the level of the water
in it is the same as that in the aspirator, and D is then closed. At the
beginning of the experiment a test should be made to see that all con-
nections are airtight ; this can be done by seeing that the insertion of a
glass plug at F causes the flow of water to stop.

HYGROMETRT 181

we neglect the fact that the air in the aspirator is saturated with
aqueous vapour at Tw, while that outside contains an unknown

amount. The neglect of this fact simplifies the calculation, but
may introduce an error of 1 or 2 per cent, (see p. 182). Then w

273 -4- T
grs. is the weight of aqueous vapour in V . 27Q _i_ TA ^res-

The weight of aqueous vapour which would saturate this volume
of air at TA° C. can be obtained directly from Regnault's tables,1
or by repeating the experiment with saturated air, which may be
done by drawing the air through tubes containing pieces of glass
tube moistened with water before it comes to the drying tubes.

273 4- T

If W is the weight of water which saturates V . ^^ — —r^A

&i& -+• i w

litres of air at TA, the relative humidity of the original air is

, w
simply -yf.

Although the above method is good enough for many purposes,
the following method of calculating the pressure of the aqueous
vapour from the results of the experiment without making
simplifying assumptions should be carefully considered, not only
for its own sake but because it illustrates the methods adopted
in many operations with gases.

Consider the air which has passed through the tubes. It starts
at a temperature TA° C., and contains aqueous vapour at a pressure
/ which we are trying to find. It gives up its moisture and comes
into the aspirator where it takes up the temperature of the water
Tw° C. and becomes saturated with aqueous vapour at this
temperature. Its volume is then V litres (which is measured),
and we require to know its original volume V. Let H be the
barometric pressure, and p the saturation pressure of aqueous
vapour at Tw°. In the aspirator, the partial pressure of the air
is H — p, the remainder of the total pressure, p, being due to
vapour. Before it came in, the partial pressure was J.—J. Thus
V litres of air at TA° and H— / mm. have become V litres at
T,,0 and H— p mm. Thus by the ordinary Boyle and Charles
Laws we have

v,_v H^p 273 + TA

' H - / ' 273 4- T. ' ^ * '

1 See Kaye and Laby's Tables of Physical and Chemical Constanta,
Third Edition, page 39. Published by Longmans. Green & Co.

182 HEAT

As we do not know /, we cannot determine V by this equation
alone.

But we have seen on page 174 that at temperatures and pres-
sures such as we are now considering, water vapour may be
regarded as obeying the gas laws, and we can calculate its density
at any pressure just as we should in the case of a gas. Now it is
found that the density of water vapour is -622 times that of air
at the same temperature and pressure.1

Now 1 litre of dry air at 0° C. and 760 mm. weighs 1-293 gms.
. * . 1 litre of dry air at TA° and / mm. weighs —

/ 273

X 760 X 273 + TA
. • . 1 litre of water vapour at TA° and/ mm. would weigh —

f 273

0-622 X 1-293 X ^ X 273 + TA gm.

But the actual weight of the water vapour in V litres was
w grams.

/ 273

.-.«,= 0-622 X 1-293 X -X X V

= 0-8 X X

273

2?3

+T x V. Substitute for V by

means of (1) and we have

/ 273 273 -f TA H-« ,,

,... _ fi.Q v \/ _ v _ X — . "X V

X 760 X 273+TAX 273 + T. H=7

/ 273 H-p

* 760 X 273 + T,, X H-/
which gives the value of /, since H, w, V, Tw and p are known.

It will now be seen that the simplifying assumption mentioned

H— p
above, was the neglect of the factor TT — j ; in most cases

/ is not likely to differ from p by as much as 12 mm., which is
less than 2 per cent, of 760 mm.

Rate of Fall of Clouds and Mists. — When a drop of water
is falling through air, it experiences a resistance which increases
with the velocity. When the velocity becomes so great that the

1 This is an experimental result due to Regnault. It may be noticed
that since the Molecular Weight of H,O is about 18, we should expect

Q

the vapour density with respect to air to be about -- — -626.

HYGROMETRY 183

resistance is equal to the weight of the drop, no further increase
takes place, and the drop falls with a uniform velocity called the
limiting velocity. This limiting velocity is proportional to the
square of the radius of the drop ; x for a drop of radius *01 mm.
the limiting velocity is something like 1 cm. per sec. A dense
fog is made up of a large number of very small drops ; so also is
the sort of cloud seen at fairly high elevation. The rate of fall
of these is excessively slow. A so-called Scotch mist is made up
of larger drops, and can be seen to have an appreciable rate of
fall. The rate of fall of large rain drops (e.g. a thunder shower)
is visibly greater than that of small.

Vapour Pressure at a Curved Surface — Condensation Nuclei.
— The saturation pressure at a curved surface differs from that
at a flat ; for a convex surface it is greater, and at a concave it
is less. The difference increases with the curvature of the sur-
face. Thus a very small drop of water, where the curvature of
the surface is great, would tend to evaporate in a space where the
pressure of the vapour was the ordinary saturation pressure.
This accounts for the fact, demonstrated by Aitken and investi-
gated in detail by C. T. R. Wilson, that in air which is free from
small dust particles it is difficult for a cloud to form, even though
the pressure of vapour is greater than the saturation pressure.
Wilson, by suddenly cooling air saturated with aqueous vapour,
showed that if the air were carefully freed from dust, a degree of
super-saturation of something like eight tunes the normal was
required to form a cloud, although a few large drops were formed
if the supersaturation was about four times the normal. With
ordinary air a dense cloud is formed with a smaller degree of
supersaturation. The function of the dust particles is to provide
nuclei for the drops to form on ; otherwise it is difficult to start
a drop since its radius must be indefinitely small to begin with,
and therefore its curvature indefinitely great. For this reason
the formation of dense fogs (i.e. fogs where there are large num-
bers of very small drops) is much facilitated by the presence of
particles in the air ; towns with a smoky atmosphere like Sheffield

1 Stokes showed that the resistance at any velocity is given by 6wr/j.v
where /* is the coefficient of viscosity of the air. If u is the limiting

velocity, 6irr/xu must equal the weight of the drop -^prr8 pg (neglecting

buoyancy of air). Thus 6*r>u =» - irr^pg or u = — • £? • r*.

3 9 fj.

184

HEAT

are subject to dense fogs, and a generation or two ago London
used to provide very fair specimens. Delayed boiling is also an
example of the influence of curved surfaces on vapour pressure.
Consider a small bubble of vapour in the liquid. Since the inside
surface is concave, the pressure of vapour will not be equal to the
atmospheric unless the temperature is above the normal boiling
point. Thus it is difficult for the bubble to form unless it has
something to form on. The influence of surface tension also
comes in.

In the absence of anything on which to form the temperature
may rise considerably above the boiling point, but if a bubble
does form it grows with great violence, since the pressure in-
creases with the size of the drop owing to diminished curvature.
Lord Kelvin's proof that the vapour pressure depends on
the curvature of the surface.

Let Fig. 83 represent a dish of water in which is placed a
narrow tube open at each end. Let the
whole be placed under an exhausted bell
jar, so that the space inside contains only
vapour. Owing to surface tension, the
water rises to a height h in the tube, h being
inversely proportional to the bore of the
tube. The surface of the water in the tube
will be concave. Now the pressure of the
vapour in contact with the water in the
dish will be p, where p is the saturation
i pressure.

Since, however, the vapour has weight
like any other fluid, the pressure at a

height h in the vapour will be less than p by an amount
equal to the pressure due to the weight of a column of vapour
of height h. The surface of the liquid in the tube is there-
fore in contact with the vapour at a lower pressure than p ;
if then curvature makes no difference to the vapour pressure,
water would tend to evaporate until the pressure in its neigh-
bourhood became p. But this would make the pressure at the
surface of the dish greater than p and condensation would take
place. Thus water would continually evaporate from the tube
and more would rise from the dish to maintain the height h.
Thus we should have continuous circulation, and " perpetual
motion." As this does not occur the water in the tube must be

Fia. 83.

HYGROMETRY 185

in equilibrium with the vapour at the lower pressure. In the case
of a liquid with a convex surface, e.g. mercury, the liquid in the
tube would be below that in the dish, and the same argument
would show that the vapour pressure at the convex surface was
greater than that at the flat. To obtain a quantitative relation,
let a = density of the vapour. Then the difference between the
pressure at the two levels is hga dynes per sq. cm.1

Now if T is the surface tension of the liquid, r the radius of the
tube at the top of the liquid column, and p the density of the
liquid, then if the angle of contact is 0° nrzphg = 2nrT *
2T

°Th=pir'

If we regard the surface of the water as approximately semi-
circular, r is also the radius of curvature of the surface.

Thus the difference of pressure at the two surfaces is

9T1 rt 1

**-*>.- --*I.j.T.

1 Strictly speaking a would be greater at the bottom than the top
owing to the compressibility of the vapour.

2 Strictly irrzgh (p-<r), since pressure on the top of the column is hga
lees than on the surface of the liquid.

CHAPTER XII

TRANSMISSION OF HEAT

CONDUCTION

In the two preceding chapters we have seen something
of the nature of heat and its relation to work ; we have now
to consider the ways in which heat can pass from one place
to another, and the conditions which govern its transmission.

Let us suppose we have a thick metal vessel (Fig. 84),
containing water, and we place it on the flame of a bunsen
burner. Consider, first, what happens in the metal. The
outside portion of the bottom, which is in contact with the
flame, becomes hotter ; the layer next to this receives heat
from it, and so on until the inside of the metal in contact with
the water becomes heated and begins to communicate heat
to the water. At this stage there is a progressive fall of
temperature from the outside portion in contact with the
flame to the inside, which is just warmer than the water.
The metal continues to rise in temperature, but the rise is
soon checked ; the beat communicated to the outer layer is
carried off as it is supplied.1

At this stage it is still true that the side of the metal in
contact with the flame is hotter than the side in contact with
the water.

Heat is passing into the water through the metal ; it is
said to be transmitted by conduction. Each portion of the

1 As, however, the water gradually becomes warmer the temperature
of the metal rises gradually also until the boiling-point is reached, when
the temperature of each part of the metal remains stationary. Heat
continues to be supplied to the water and is used up in turning water
to steam (see page 195).

186

TRANSMISSION OF HEAT

187

metal along the direction of flow receives heat from a part
at a higher temperature and passes it on to one at a lower.

The distribution of the heat to the different parts of the
water is chiefly brought about by another method. The
water heated by the metal expands, and, becoming less
dense, rises to the surface, as indicated by the arrows, while
colder water flows in to take its place and become heated in
turn. A circulation is started, which can be made evident
by putting in a few pieces of damp
sawdust or bran. Thus the heat
travels to different parts of the vessel
by the actual movement of portions
of the water conveying heat with
them. This process of distribution
of heat is therefore spoken of as
convection.

There is, however, a third method
by which heat can be transmitted from
one place to another ; this method
is altogether different from those we
have considered. If we stand in the
sunshine, we are conscious of a feeling
of warmth ; it is not a case of the air
being warmed and passing on heat to
us, for if a cloud passes between us

and the sun, we are immediately aware that the heating
effect has stopped. Moreover, the effect is very noticeable
even when the temperature of the air is below freezing-point,
e.g., at high elevations in Switzerland or Norway one can
often sit comfortably in the sunshine with one's coat off
although the snow is hard and the air below freezing-point.

This is a case of the transmission of heat by radiation, and
the essential point is that the transmission takes place without
raising the temperature of the intervening medium. This
essential point is well illustrated by a modification of the
well-known effect of a lens or " burning glass," by which the
sun's radiation is made to light a pipe "or set fire to paper ;

j

1

tr

i|

N

tt

^

*

tt

if

84.

188 HEAT

a lens of ice is made by means of a mould, and the effect of
setting fire to paper can still be produced, although the
transmission must have taken place through a medium at a
temperature not greater than 0° C.

A large amount of the heat received when one stands before
a fire is transmitted by radiation ; very little by conduction
through the air.

Thus there are three ways in which heat can be transmitted
from one place to another : —

Radiation. — The transmission takes place without raising
the temperature of the intervening medium. It can take
place most readily across a vacuum.

Conduction. — The temperature of the intervening medium
is raised, but there is no appreciable 1 movement of portions
of the medium along the direction of flow of heat.

Convection. — Heat is conveyed by appreciable portions of
the medium moving along the direction of the flow of heat.

"jTnese three methods must now be considered separately
in greater detail.

CONDUCTION

Conductivity of Solids. — It is easy to show that some
solids conduct heat much more readily than others. For
example, if a silver teaspoon or a short length of thick copper
wire has one end immersed in a vessel of boiling water, the
other end rapidly becomes too hot to hold ; under the same
conditions a glass rod or tube can be held for any length of
time without becoming particularly warm. When glass
tubing is being worked in the blow-pipe flame it is possible
bo take hold of a part of the tube only an mch or two from
the flame.

We therefore say that silver and copper are good conductors
of heat, while glass is a bad conductor. The thermal con-
ductivity of a substance has, however, a precise meaning, and
can be expressed as a number.

1 The word " appreciable " is put in so as not to exclude molecular
movement.

CONDUCTION 189

Definition of the Thermal Conductivity of a Sub-
stance.— Consider a plate of the material of thickness t.
Let one face of the plate be maintained at a temperature 0,
and the opposite face at 02. Then heat will be flowing
through the plate, and experiment shows that for any given
material, the quantity of heat flowing through the plate per
second is proportional to the temperature difference 6^— 0,
and inversely proportional to t. It is also, of course, pro-
portional to the area of the plate, and depends on the material.

The thermal conductivity of the material is defined as the
amount of heat per second which flows across unit area of a
plate of unit thickness when the difference of temperature of
the opposite faces is maintained at one degree. Thus if K

were the thermal conductivity of the material, the amount

/\ r\

of heat flowing per second across unit area would be K . — ?.

r\ /\

We may call — the temperature gradient along the

direction of the flow of heat, and we can then get a shorter
and more general definition of K as follows : —

The thermal conductivity of a substance is the amount of
heat flowing per second across unit area perpendicular to the
flow when the temperature gradient along the direction of
flow is one degree per unit length.

Determination of the Thermal Conductivity of a
Metal by Searle's Apparatus. — A solid bar of the metal (Fig.
85), about 5 cm. diam. and 20 cm. long, is fitted at one end
to a circular chamber A, through which steam can be passed.
Near the other end a coil of thin copper tubing, B, is soldered
to the bar. ( The ends of this tubing are connected to copper
cups. Cold water circulates through B. The bar with its
fittings is surrounded by thick layers of felt packed in a
wooden box, made in two halves ; in the figure one half is
removed. Steam circulates through A, and the rate of flow
of the cold water through B is kept steady. This water
receives heat from the bar, and the amount of heat received
per second is determined by finding the weight of water

190

HEAT

flowing in a given time and reading the difference between
the temperature of the water when entering and leaving B.
The thick felt packing prevents any appreciable loss of heat
from the bar except that which is given to the circulating
water ; a steady state is therefore reached in which the flow
of heat down the bar is equal to that received by the water

Fio. 85.

When this state is reached there is a temperature gradient
down the bar, and since heat enters the bar at the end in
contact with the steam chamber and leaves it at B and no-
where else, it follows that the amount of heat which flows
per second across any section between B and A is the same.

The temperature gradient can be measured at any con-
venient place, and is determined by the difference between
the readings of the thermometers Tj and T2 placed in copper
tubes C and D brazed into the bar at a known distance apart.1

1 It would not be at all accurate to assume that the hot end of the
bar was at 100° C. and the cold end at the mean temperature of the
water. In the case of a good conductor like copper the temperature

CONDUCTION 191

In carrying out an experiment it is necessary to wait until
the readings of all four thermometers are constant. The
following figures will explain the method of calculation : —

Reading of T3 ...... 23° C.

,, T4 ...... 14-7° C.

Rate of flow of water . . . 205 gm. per minute

=3417 gm. per second

Diameter of bar . . * . . . 3-80 cm.
Reading of Tl . . . . . 79° C.

„ Ta . . . . . . 51° C.

Distance apart of Ta and Ta . . . .10 cm.

Rate of flow of heat along the bar=(3417) X (8-3)

=28-36 calories per sec.
Area of cross section of bar=^r2=7i (1-9)2 = 11-34 sq. cm.

00 Of*

/. Rate of flow per sq. om-=if|j=2-5<) cals. per second.

28
Temperature gradient = -— = 2-8° C. per cm.

Comparison of Conductivity. — A rough comparison of
the conductivities of different metals can be made by an
experiment similar to that of Ingen Hausz in 1789.

A number of bars of the same length and thickness, but
made of different materials, project through tubulars in the
side of a rectangular metal vessel (Fig. 86). A number of
marbles are stuck by means of small pieces of wax to the under
side of each bar. The vessel is then filled with boiling water,
arid the water is maintained at the boiling-point by means
of a burner ; a doubled-walled screen, K, prevents the wax
being melted by direct radiation from the burner. As heat
is conducted along the bars, some of the marbles fall off
owing to the melting of the wax ; the experiment is continued
until no more fall off. It will then be found that the bars
have lost different numbers of marbles ; in some only the
first marble or two nearest the hot water have fallen, and in

of the hot end is considerably below 100° C. and that of the cold end
considerably above that of the circulating wnt*r.

192

HEAT

others considerably more. A good selection of metals to use
would be copper, aluminium, zinc, iron, lead, bismuth. The
effect will be much the greatest with copper and least with
bismuth.1 If a brass bar is included it will be found to be
nothing like so good a conductor as copper, being about equal
•to zinc.

It is important to notice that it is the final state of the bars

which enables us to de-

cide which
conductor,
first begins
temperature

is the best
When heat
to flow the
at points

FIG. 86.

along a bar rises ; the rate
of rise depends not only
on the rate at which heat
is flowing, but also on the
Specific Heat of the metal.
Thus, the initial rate of
rise of temperature along
a bar of metal of low
Specific Heat might be
greater than in the case
of a metal of high Speci-
fic Heat, even though the

latter were the better conductor. The first marbles to drop
off do not necessarily come from the best conductor. But
as the temperature of the bar rises it begins to lose heat to
its surroundings, partly by radiation and partly by convection
in the air. A steady state is finally reached when the rate of
loss of heat from this cause just balances the flow of heat by
conduction. Thus the amount of heat which flows across any
section A of the bar is equal to the amount emitted from the
surface of the bar beyond A. If much heat is flowing across
A the temperature of the bar beyond will have to rise higher

1 The bars should be fairly thick, e.g. about 1 cm. diameter, otherwise
the worst conductors, lead and iron and possibly even zinc, will not lose
any marbl**. at all.

CONDUCTION 193

than if only a little flowed, in order to get rid of this quantity.
Thus, if the bar is a good conductor, the final steady tempera-
ture at any point A will be higher than at a corresponding
point of a poorer conductor.1 The temperature of any point
of a bar is, of course, lower the further the point is from the
hot end ; heat Would not flow from A to a point beyond it if
both parts were at the same temperature.

This method will show whether one metal is distinctly a
better conductor than another, but, unless great care is taken
that the surfaces of all the bars are precisely similar as regards
their power of emitting heat, it does not give the ratio of the
conductivities with any approach to accuracy. If the sur-
faces are similar, it can be shown that the conductivities are
approximately proportional to the square of the distances
from the hot end at which the temperature just attains the
melting-point of the wax (or any other given value the same
for each bar).

The absolute value of the thermal conductivity of a few
substances are given below ; they represent approximate
values at ordinary temperatures.2

Silver . . . ; . . -97

Copper ... . -92

Aluminium . ^ . . '50

Steel -11

Lead . ... . . -08

Brass -26

Earth's Oust . . . about -004

Glass . . . . about -0025, depending on the

kind of glass.
Wood. .... about -0005, depending on the

kind of wood.
Cork . . . ... -00015

Cotton Wool * ... . -00004

1 This assumes that the surfaces of the two bars are similar. The
rate of loss of heat from a surface depends on the temperature and also
on the nature of the surface: see page 214.

2 More accurate values of the thermal conductivities of many sub-
stances are given in Kaye and Laby's Tables of Physical and Chemical
Constants.

194 HEAT

Thermal and Electrical Conductivities.— Silver is the

best conductor of heat and is also the best conductor of
electricity, with copper a close second. It is found that for
metals and alloys, the ratio of the thermal to the electrical
conductivity is approximately constant at any given temper-
ature ; this is known as the Weidemann Franz Law. There
are, however, no substances known which act as insulators
to heat in anything like the same way that such substances
as glass or quartz do to electricity.

Conduction of Heat through Liquids. — When a liquid
is heated from below, transmission of heat through the liquid
takes place mainly by convection. If, however, the liquid
is heated at the top, the transmission of heat downwards
must take place by conduction, and it is found that the con-
ductivity of most liquids (except mercury) is low. This
may readily be illustrated by taking a boiling-tube containing
water, and placing in it a piece of ice wrapped in wire gauze
so that it sinks to the bottom. If the water be now heated
near the top by a burner, the water at the top can be made
to boil briskly for some considerable time without melting
the ice. The conductivity of water at ordinary temperatures
is about -0014, i.e., about half that of glass ; 1 other liquids
mostly have conductivities a good deal less than this.

Conductivity of Gases. — The conductivity of gases is
very low ; in the case of air it is somewhere about -00005 ;
hydrogen has a conductivity six or seven times that of air.
The low conductivity of air is the reason why some materials
such as blankets and woollen materials are such bad con-
ductors of heat. They owe their properties chiefly to the
numerous small air spaces in the texture of the substance.
Woollen material pressed out under great pressure loses
very much of its power of diminishing the transmission of
heat. In the same way cotton is not a bad conductor in the
ordinary way, but when made up into the form of cellular
material it forms a substance which is quite suitable for use
instead of woollen stuffs for clothing purposes.
1 See note at the end of the chapter.

CONDUCTION 195

Cotton wool is another example of the effect of enclosed air.

Transmission of Heat by Conduction. Some Practi-
cal Considerations. — Returning to the case of the metal
vessel considered on page 187. As soon as the outside is
heated, heat begins to flow through the metal, and finally a
steady state is reached at which the heat which flows through
the metal is equal to that supplied to it by the flame. At
this stage, however, the temperature of the outside of the
vessel is by no means equal to that of the flame. The fact
that heat is taken from a portion of the flame causes the
formation of a film of relatively cool gas over the surface of
the metal in contact with the flame ; this film remains.

The heat from the flame has to pass through this film, and,
since gas is a bad conductor, there must be a very consider-
able difference of temperature between the sides of the film,
and this difference will be greater the greater the amount of
heat flowing through per unit area.1 This film of gas is a
great problem to engineers, and various devices have been
designed to minimize its effects.

A very simple illustration of this point is furnished by the
well-known experiment of boiling water in a paper vessel.
A box is made of paper, not too thick, and so long as the
flame does not touch the paper anywhere above the water
line or at any place where there are several folds, the water
is boiled without charring the paper.

Another experiment which illustrates this point, and also
the different behaviour in the case of good and bad con-
ductors, is the following : —

A stout bar is taken, one half of which is brass and the other
wood. A piece of fairly thin white paper is wrapped tightly
once round the bar (the edges may overlap an inch and be
secured with gum). The bar is held with its centre in the

1 A similar effect is produced in the case of the experiment on page
190. When the end of the copper bar is in contact with steam, a film
of moisture forms, and the temperature of the face of the copper is
lower than that of the steam. In the case of a good conductor like
copper the difference of temperature between the sides of the film may
be greater than that between the ends of the bar.

196 HEAT

flame of a bunsen burner. It will be found that the paper in
contact with the wooden part is charred, but not that in
contact with the metal. A sharp line will mark the junction
of the wood and metal.

A comparatively small rise in temperature of the metal in
contact with the paper results in a considerable flow of heat
to the colder parts. There is therefore a considerable flow
of heat across the film of gas between the paper and the rest
of the flame, and so, as we have seen, there must be a, large
temperature difference between the flame on one side of the
film and the paper on the other.

The wood in contact with the paper, being a poor conductor,
will rise to a very considerable temperature before any large
flow of heat to the colder parts takes place. The paper is
therefore raised in temperature sufficient to char it.

Scale in Boilers. — Owing to the " hardness " of water a
deposit known as scale tends to be deposited on the sides of
boilers (similar to the "fur" in kettles). Serious conse-
quences may occur if this is allowed. If scale is deposited
on the furnace tubes, not only is the heating surface less
effective, owing to the bad conductivity of scale, but owing
to this bad conductivity heat does not flow through quickly
until a large temperature gradient is set up, and so the part
of the furnace tube in contact with the scale becomes raised
to a higher temperature than the rest, in the same way as the
paper in contact with the wooden part of the compound bar.
The excessive expansion of this part sets up stresses, and
may seriously weaken the boiler.

Action of Wire Gauze. The Davy Safety Lamp. — If
a piece of fine-meshed wire gauze be placed over a bunsen
burner (not too far above it), and the burner be lighted below
the gauze, the flame will not pass through to the upper side
(Fig. 87a) ; if it be lighted above the gauze, the flame does
not pass down to the under side, but remains as in Fig. 876.

The part of the gauze in contact with the flame loses heat
by conduction to the colder parts so rapidly that its tempera-
ture is therefore not nearly so high as that of the flame, and

CONDUCTION

197

FIG. 87.

is, in fact, well below the ignition-point of coal gas. (In the

case of a burner supplied with hydrogen, which has a lower

ignition-point, the flame does pass through the gauze.) Thus

any coal gas above

the gauze in a or

below it in 6 does

not become ignited.

It can, however, be

ignited by bringing

a light to it, and if

this is done the

flame continues to

burn above and

below the gauze.

This principle was made use of by Humphry Davy in the
lamp he designed for use in coal mines. The flame of the
lamp is surrounded by a cylinder of wire gauze. Thus if the
lamp happens to be taken into an atmo-
sphere containing some coal gas, the gas
outside the lamp will not be ignited ; some
of it will enter the lamp and it can be seen
burning, forming a " cap " to the flame.
By the look of the flame an experienced man
can tell whether combustible gas is present
and give a rough idea of the proportion of
it.

In connection with conductivity, one
practical point may be noticed. It will be
seen from the table on page 193 that the
conductivity of the earth's crust is quite
considerable. In sleeping out, therefore, the
loss of heat to the ground tends to be greater
than the loss in other directions, and if only limited cover-
ing is available, the greater portion of it should be under
rather than over the sleeper.

Note on the Experimental Determination of Thermal Con-
ductivities.— The earliest successful numerical determination

FIG. 88.

198 HEAT

was that of Forbes, 1850. He used a bar heated at one end as
in the work of Ingen Hausz, but measured the temperature at
different parts of the bar by means of thermometeis inserted in
cavities ; contact being made by filling the hole with mercury or
fusible metal.1

When a steady state is reached, the heat flowing across any
section of the bar is equal to that lost from the surface of the bar
beyond. Since heat escapes from the sides, the amount flowing
across any cross section diminishes as the distance from the hot
end increases. The temperature gradient becomes therefore
progressively less.

The difficulty is to determine the rate of loss of heat from the
sides of the bar. This was done by a second set of experiments
to determine the rate of cooling when bars cf the same material
and section as the original bar were heated and then left to cool.
The method was very laborious, but reasonably consistent results
were obtained.

Modifications of the bar method have been used in modern
work on metals. The rate of flow of heat, however, is measured
more directly. One way, used by Callendar and Nichols, is to
measure the heat which has passed through the bar by a calori-
metric method (Searle's apparatus is an example of this method).
Another way, used by Lees, is to supply the heat electrically
and deduce the amount supplied.

Conductivity of Bad Conductors. Lees' Disc Method." —
In outline the method is as follows. The substance was in the
form of a disc 2 or 3 mm. thick and some 4 cm. in diameter.

It was placed between two simi-
lar discs of copper, A and B
(Fig. 89). Against the face of
A is a flat coil of insulated wire
which could be heated by pass-

_ ing a current through it. Above

this is another copper disc D. A

thin layer of glycerine was used to make good thermal contact
between the discs. The whole pile was varnished all over to give
it a uniform surface, and was suspended in a constant -tern -

1 Despretz had already used this method to verify the laws of con-
ductivity.

* Lees, Phil. Trans., 1898.

CONDUCTION 199

perature enclosure. A current was sent through the heating
coil, and the temperature of the copper discs was determined
by thermo-couples.

After a time a steady state was reached, the heat generated in
the coil passing out to the surfaces of the pile, and so by radiation,
conduction, and convection to the enclosure. Since copper is a
very good conductor, the temperature gradient is negligible in the
copper compared to that in the substance ; the temperature of
each disc is therefore taken as constant and given by its thermo-
couple ; the temperatures of the faces of the disc of substance are
given by those of A and B respectively.

Since the temperatures are not high, Newton's law of cooling
can be assumed ; the heat lost per second per unit area of any
portion of the surface pile will be some constant h multiplied by
the excess of temperature of the surface over that of the
enclosure.

If T, is the temperature of D and Al area of its exposed surface
-*- 2 »» » »> •**. and A.2 >» ,, ,,

T3 „ „ „ B and A3 ,, „ „

and A4 the area of exposed surface of the substance the total
heat lost per second is

* + T$-TJ-

where T is the temperature of the enclosure.

But this total is equal to the rate at which it is supplied elec-
trically, and so is known.

This therefore gives h.

Knowing h, the amount of heat passing through the disc per
second is got ; it may be taken as the mean of that passing out,
i.e. ^A3(T3— T), and that passing in, i.e. ^A3(T3— T) + the loss

/T -I- T \

from the sides of the disc, which is h&A -*-g — 3 — TJ.

Thus the heat passing through per second is known.
The temperature gradient is thicj^ % disc

Thus we have the heat flow and the temperature gradient, and
so the conductivity.

The above is a mere outline, and corrections are necessary.

Conductivity of Liquids. — Lees applies the method to
liquids, the liquid being confined between copper discs by
means of an ebonite ring. The rate of flow of heat was deter-

200 HEAT

mined by including in the pile a sheet of substance of known
conductivity.1

1 A good account of the methods of determining conductivity is
given in Preston's Theory of Heat, and also in Poynting and
Thomson. A very complete account of the subject is given in A
Dictionary of Applied Physics, Vol. I, published by Macmillan,

CHAPTER

TRANSMISSION OF HEAT BY CONVECTION— SOME
PRACTICAL EXAMPLES

We have already seen the large part played by convection
in the transmission of heat through liquids ; in the present
chapter we shall consider some important examples.

Circulation in Boilers — Use of Cross Tubes. — In order
that heat from the furnace may
be communicated to the water
as rapidly and evenly as possible
it is necessary to have an
adequate circulation of the
water. The ordinary convection
currents set up by the action
of the furnace provide a certain
amount, but the circulation is
much improved by fitting a few
cross tubes, as shown in Fig. 90.
These tubes are inclined at an
angle of about 45° to the hori-
zontal, and the water in them
being heated by the hot gases from the furnace becomes
less dense and rises, more water flowing in from below. A
very brisk circulation is set up in this way.

Hot -water System of Heating Buildings.1 — The boiler,

1 The account in the text gives an idea of general principles, but
there are several ways of applying them, and a number of important
practical points arise. A standard work on the subject is Heating by
Hot Water, Jones Walker, published by Crosby Lockwood & Sons.

An instructive article on heating and ventilating buildings is to be
found in the Journal of the Royal Sanitary Institute, February, 1910, in
which there are a number of useful diagrams.

2QL

Furnace tube

FIG. 90.

202

HEAT

usually of the shape shown in Fig. 91, is placed somewhere
in the basement. A pipe leads out from the top of the boiler
to the highest point of the system at A. From there it goes
to the room or rooms to be warmed and returns to the lower
part of the boiler at B.

The boiler and system of pipes are full of water, and in
order that they may remain full and yet allow the water to
expand a small open tank D is arranged at some level above

Ux.

FIG. 91.

A, and communicates with the lower part of the boiler at C.
The water in the upper part of the boiler and the pipe
immediately above it is heated and so becomes less dense ;
it is therefore displaced by the colder water which flows in
at B. The hot water flowing along AL gives up some of its
heat to the room and so becomes colder ; the circulation
once set up is thus continued. Heat is supplied to the water
in the boiler and some of this heat is conveyed by the cir-
culation and given up to the rooms.

In order to increase the heating surface of the pipes a
number of radiators (groups of short vertical pipes) may be
included in the part AL, or the pipe may be arranged with

CONVECTION

203

coils. It may be noted that the heat is delivered to the room
from the surface of pipes and radiators mostly by convection
currents in the air and radiation plays a comparatively small
part. Where there are a considerable number of rooms to
be heated the pipe branches at A to form a number of circuits,
or there may be separate upright pipes from the top of the
boiler. Care is necessary in designing the system, otherwise
there is a much greater circulation in one route than another.
Circulation is much impeded if air collects in the system,
so that an air-escape valve must be fitted at the highest
point (sometimes merely a vertical pipe open at the top,
which must be above D) and should
also be fitted at the top of each
radiator.

The rate of circulation depends,
among other things, on the vertical
height from A to B ; if the avail-
able height is small and the hori-
zontal extent of the building con-
siderable, convection currents are
insufficient and a forced circulation
is provided by a pump.

Water-cooled engines on a car
provide an example of a very
similar set of operations. A large
amount of heat is communicated to the cylinder walls, and
scfme of this must be got rid of if the engine is not to over-
heat. The cylinders are provided with water-jackets A (Fig.
92) connected to a radiator R ; the latter is essentially a
tank having a top and bottom portion connected by thin
metal vertical pipes between which air can pass freely as
the car moves. The circulation of air is often aided by a
fan driven from the engine. The connection between water-
jacket and radiator is by the pipes B and C. B must start
from the highest part of the jacket and slope upwards to
the top of the radiator and C should slope downwards from
the lowest portion of the jacket to the bottom of the radi-

FIG. 92.

204 HEAT

ator. The radiator and jacket are full of water ; the water,
heated by the cylinder, becomes lighter and is displaced up
B to the top of the radiator. Circulation is established, the
heated water coming to the top of the radiator being
cooled as it passes down the tubes. There is thus continual
transference of heat by convection from the hot cylinder to
the radiator tubes and thence to the air. This system of
circulation is spoken of as the thermo-syphon principle, and
it is to some extent self -regulating ; the greater the develop-
ment of heat in the cylinders the -more rapid is the circula-
tion. In some cases this does not provide sufficiently rapid
movement of the water and a circulating pump is provided,
driven off the engine ; this applies particularly to aeroplane
engines, where the problem is further complicated by the fact
that the aeroplane is by no means always in a horizontal
position.

Domestic Hot Water Supply. — Fig. 93 is a diagram
showing the ordinary arrangement of the hot water supply
of a house. The essential parts are the hot water cistern C,
and the boiler B ; the latter is placed at the back of the
kitchen fire and must be below the level of the cistern. The
water in B, heated by the fire, rises up the pipe E into the
upper part of the cistern, and colder water flows from the
bottom of C down the pipe F into B, and becomes heated
in its turn. Thus portions of heated water are constantly
passing in at the top of the cistern and the colder water from
the bottom passing to the boiler : more and more hot water
accumulates at the top of the cistern until ultimately (if no
water is drawn off) the whole of the contents become hot
and then the circulation is less rapid. A pipe H near the
top of the cistern leads to the hot water taps. D is a tank
of cold water which communicates with the lower part of
the cistern and provides the necessary pressure to drive out
the hot water when the taps are opened ; the colder water
flowing in. As the water flows out of D the float K descends
and opens the valve L which admits more water from the
supply until the level is restored. The open pipe G is a

CONVECTION

205

safety outlet in case the water in the cistern becomes so hot
as to boil.

In the above examples we have been concerned with con-
vection currents as a means of transferring heat from one
place to another : there are, however, many cases where
the interest from a practical point of view is centred in the
actual movement of the liquid or gas rather than in the
^ansference of heat. Examples of such cases are the action

From Cold ,
Water Supply

Fio. 93.

of chimneys, ventilation of buildings and atmospheric
movements such as the Tr^de Winds.

Action of Chimneys. — The gases in a chimney above a
fire are hotter than the air outside it, and consequently have
expanded and become less dense.1 The pressure at the base
of the chimney due to these lighter gases is less than that of

1 The gases in the chimney will include the products of combustion
some of which, e.g. CO2, are heavier than air at the same temperature.
On the other hand some, e.g. water vapour, are lighter, and the largest
part (residual nitrogen) about the same density. The great bulk of
the products at the high temperature will be considerably lighter than
the outside air, and even CO2 does not require to be at as high a tem-
perature as 200° C. to be lighter than air at 15° C. and the same pressure.
Under natural draught 1 Ib. of coal requires at least 24 Ib. of air, so
that 24 Ib, of air only make 25 Ib. of products of combustion.

206 HEAT

the air at the same level ; the latter therefore flows into the
chimney above the fire and through the grate displacing the
lighter gas. This air becomes warmed in turn, and so a
circulation is set up. This circulation is of the utmost import-
ance in getting the fire to burn. For example, it sometimes
happens that when a domestic fire is first lighted difficulty
is experienced in getting it to start : the fire will not " draw "
and shows a tendency to smoke into the room. Warming
the air in the chimney (e.g. by burning paper) will often
start the circulation in this case, and the fire will then burn
satisfactorily.

When the fire in a room is well alight, the quantity of air
passing up the chimney is very considerable — much more than
is required for combustion. Fresh air comes into the room
by any available openings, e.g. at the top and bottom of the
door and by the windows (if the latter are shut they are not
likely to be air-tight and air gets in through the chinks).
In this way a fire is an important ventilating agent, and its
merit is that air does definitely go out of the room and fresh
air comes in ; it is not merely a case of setting up circulation
within the room.

Ventilation. — In the consideration of the problem of
ventilation the properties of convection currents must be
considered not only with regard to the main movement of
the air, but also in connection with local effects. Any object
which warms the air in its neighbourhood results in an upward
air movement, and equally any cold object results in a down-
ward current. For example, large windows, even if closed,
mean a surface which in cold weather is colder than the rest
of the walls, so that the air inside tends to be cooled and
quite a considerable down-draught of cold air may be set
up. This effect is so noticeable in some cases, e.g. churches,
that pipes have to be arranged near the wall to warm this
air and prevent the down-draught. Again, each person to
some extent produces his own ventilation current ; he warms
the air in his neighbourhood, and the air he breathes out is
warmer (and contains more water vapour but also more

CONVECTION

207

carbon dioxide) and so tends to rise out of the way. If it
were not for this effect any one sleeping in bed would tend
to surround himself with a layer of unbreathable air. Thus
vitiated air tends to rise to the upper part of the room, and
the problem is to arrange for its removal and replacement.

The effect of an open fire in renewing the air of a room
has already been mentioned ; if, however, the entering air
merely comes in at doors and windows it will tend to sink
and travel as a cold current near the floor direct to the fire-
place, leaving the upper air undisturbed. A common arrange-
ment intended to assist
the displacement of the
upper air is to bring in
the entering air by a
Tobin tube (Fig. 94)
delivering above the
heads of the occupants ;
if, however, the air is
cold it will still sink

FIG-

when it gets in. A more

successful arrangement is

to allow the entering air

to be warmed by passing

over a heated surface, in

which case it is more effective in sweeping out the vitiated

air, since it no longer sinks to the floor level.

A common method of getting air to enter in the case of
large halls heated by radiators is to have a row of radiators
along a wall and air inlets behind them ; the radiator causes
an upward convection current and fresh air enters through
the inlets and is warmed in turn.

In some systems of ventilation the used air is extracted
(by means of fans or by a heating arrangement in an external
flue) causing a reduction of pressure so that fresh air enters
automatically where it can ; another system (the plenum
system) delivers fresh air through inlets and causes a slight
excess of pressure so that used air is driven out ; neither of

208 HEAT

these systems can be regarded as very satisfactory, and it is
better to have two separate arrangements, one driving in
fresh air and the other extracting used air.

It may be mentioned that, considered solely from the point
of view of setting air in motion, direct heating is much less
efficient than using the heat in an engine and moving the air
by a fan.1

Trade Winds. — Convection currents in the atmosphere
play a most important part ; as an example, the Trade
Winds may be considered. Owing to the higher temperature
in the equatorial regions of the earth, convection currents
are set up, colder air flowing in from the north and south
temperate regions near the earth's surface and a corresponding
current flowing outwards in the upper air. The lower cur-
rents form the Trade Winds, which blow with great regularity.
The direction is not exactly north and south, owing to the
rotation of the earth ; the earth rotates from west to east
and the velocity of points at high latitude is less than those
near the equator. Consequently the wind from the regions
of less velocity lags behind the regions of higher velocity
and appears to come from the N.E. and S.E. respectively.

Between the N.E. and S.E. Trade Winds is the region of
the equatorial calm where the ascending current is taking
place.

1 An admirable account of the principles of ventilation is given in
Air Currents and the Laws of Ventilation, W. N. Shaw, Cambridge
University Press, in the course of which it is shown that some forms of
ventilation systems give most unexpected and rather humorous results
(from the point of view of outside observers).

CHAPTER XIV
RADIATION

Of the two processes by which heat may l>e transmitted
from one place to another, the two already considered (con-
duction and convection) have a certain resemblance ; the
whole or part of the medium between the places is heated,
and some time elapses before the operation is completely
established. But the process known as radiation is altogether
different ; it goes on most readily when there is no material
medium between the places (i.e. across a vacuum) and with
very great velocity.

That conduction and convection are not the only means
by which a hot body can lose heat is illustrated by the follow-
ing simple and rather rough experiment. The hot body is
a glass bulb blown on the end of a piece of thick-walled narrow-
bore tubing and filled with mercury
so that it is like a large thermo-
meter ; it can be made to indicate its
own temperature by the position of
the mercury in the tube.

Two metal vessels, of the same
size, shape, and material (e.g. copper
calorimeters), are provided ; the
interior of one vessel is highly
polished, and that of the other coated
with a layer of lampblack by holding
it over the flame of burning cam-
phor. The tube of the " hot body "
passes through a rubber cork fixed in
a hole hi the centre of a piece of stout

209

FIG. 95.

210

HEAT

cardboard (Fig. 95) ; the position of the cork is adjusted
so that when the cardboard is placed on the mouth of
one of the metal vessels the bulb will reach nearly to the
bottom. Two or three fine scratches are made on the tube

about 1 cm. apart and rather
nearer the upper end ; the bulb
should also be coated with lamp-
black. The hot body is heated
by placing it hi a metal cone on
a sand bath (Fig. 96) ; it is
allowed to remain there until the
mercury is some distance above
the upper mark. It is then
transferred to the polished metal
vessel, and the time at which the
mercury passes each mark noted
by a stop-watch or an ordinary
watch with seconds hands. The
hot body is then transferred to
the heater again and heated until
the mercury is above the upper
mark ; it is then placed in the
blackened vessel and the time at

FIG. 96.

which the mercury passes each mark is again noted. The
whole experiment is then repeated.

The following readings from an experiment indicate the
general effect : the first two marks were about 1 cm. apart,
but the third was only about 0-5 cm. below the second.

First Experiment.

Time of passing 1st mark
2nd

„ 3rd „

Polished vessel.

Black vessel.

m. s.

m. s.

° 25Jlm. 7s«

, i r*>2 secs.

2 io}45secs-

1 38 j37 secs'

Total 1-3=

Total 1-3 —

1m. 52 secs.

1 m. 29 secs.

RADIATION

211

Second Experiment.

Time of passing 1st mark
2nd „

», » 3rd .,

Polished,
m. s.

0 40).

1 49/lm'9s-

2 39 r 50 sees.

Total 1-3=
1 m. 59 sees.

Black.

m. s.

2 15' 37
Total 1-3=
1 m. 32 sees.

It will be noted that in every case the time taken for the
mercury to fall from any one mark to another was distinctly
greater when the bulb was in the polished enclosure than
when it was in the blackened one. Now when the mercury
is at any particular mark the temperature must be some
particular value. Hence passing from one mark to another
represents a definite fall in temperature, and since we are
always dealing with the same mercury and bulb this repre-
sents the evolution of a definite quantity of heat. Thus
the rate of loss of heat is smaller when the bulb is surrounded
by the polished vessel.

The bulb loses heat by conduction and convection through
the air, and conduction through the glass stem, cork and card-
board.

But the conditions are similar in the two cases ; there is
no reason to suppose that the lampblack surface would
increase the convection, and certainly none for supposing
it to increase conduction.

The bulb must therefore in addition lose heat by other
process involving the emission of something which can be
reflected. This part of the loss is the radiation,1 and the

1 In order that the radiation loss may be a considerable fraction
of the total heat loss, the temperature of the hot body should be fairly
high. In the above apparatus the quantity of mercury was adjusted
so that the upper mark corresponded to a temperature of about 150°
or 200° C. It is desirable to have a considerable mass of mercury ;
the bulb in the hot body used was 3 cm. in diameter, and the vessels
were thin copper calorimeters 10 cm. high and 6-4 cm. diameter
(4 in. x 2£ in.). The blackened vessel at the end of an experiment
was always perceptibly hotter than the other.

212

HEAT

something emitted is often referred to as " radiant heat " ;
this term, however, is not a very happy one, and although
we shall make use of it for the present, we shall see later
that there is an expression for it which conveys a better idea
of its true nature (page 226).

This radiant heat has many properties in common with
light ; it travels through any homogeneous medium in
straight lines and obeys the same laws of reflection as light.
In order to illustrate these properties we shall require some-
thing with which to detect the radiant heat ; for this purpose

FIG. 97.

a thermopile (page 5) is convenient,1 and in order to get
considerable deflections on the galvanometer scale it should
be fitted with a cone polished on the inside. Three pieces
of stout card are taken, and in each of them is cut a slit
about -5 cm. wide and of length equal to the diameter of the
large end of the thermopile cone. The cards are mounted
vertically on stands, and placed in such a position that the
slits are accurately in line ; this is determined by sighting
from one end.

The cone of the thermopile is placed immediately behind
the last slit, and a large copper ball, previously heated over
a gas ring, is placed on a stand in front of the first sb't (Fig. 97).
A considerable deflection of the spot of light on the galvano-

1 An air thermometer can be used in some cases, and was used in
early experiments on radiation. It is not, however, sufficiently sensi-
tive for many of the experiments here described.

RADIATION 213

meter scale will be noticed ; but if the centre card be dis-
placed sideways so that the slits are no longer in line, the
spot of light moves back.

Fig. 98 shows the arrangement for illustrating the laws of
reflection. The third screen is moved from its place in line
with the others and placed in some such position as B. A

FIG. 98.

strip of polished metal A is mounted on a stand so that it
can be rotated about a vertical axis. It is placed in line
with the first two slits, and then rotated until an observer
looking through the slit in B sees the reflection of the first
two apparently in line. The thermopile is placed behind the
slit in B, and the hot ball behind the first. A deflection of
the spot of light on the galvanometer scale is noticed, and
a rotation of A either way causes the spot to move back.1

1 The screens should not be too far apart unless the galvanometer
is very sensitive. The writer finds that with a 20 element thermopile
and a Paul reflecting unipivot galvanometer good deflections are got
with the screens about 6 cm. apart, using slits -5 cm. wide.

214 HEAT

Emissive Power of Various Surfaces. — The intensity
of the radiation from the hot object depends not only on the
temperature, but also to a very large extent on the nature
of the radiating surface. This was established by Sir John
Leslie and may readily be shown by using a Leslie cube — a
cubical tin vessel which can be filled with hot water and of
which the sides can be coated with any desired substance.
One side should be coated with lampblack so as to give a dull
black surface, another should be highly polished, while the
other two sides may be respectively roughened and painted
white. If such a cube be filled with boiling water and placed
near a thermopile, it can be rotated so as to bring each face
in succession opposite the thermopile and the deflection of
the spot of light on the galvanometer scale noticed ; care
must be taken not to vary the distance between the cube
and the thermopile as the former is turned. It will be found
that much the smallest deflection is given when the polished
face is radiating to the thermopile while the largest is given
by the dull black face ; the other two faces will give inter-
mediate deflections, the white painted surface giving the
larger. All experiments show that a highly polished metallic
surface is an exceedingly poor radiator, while the best of all
radiators is a perfectly dull black surface.

Absorbing Power of Surfaces. — When radiation falls
upon any object some or all of it may be absorbed, and the
absorbed energy then appears as heat in the ordinary form ;
it is not until it has been absorbed that the ordinary effects
of heat are manifested. The extent to which absorption
takes place depends very much on the nature of the surface.
A highly polished metallic surface will reflect most of the
radiation which falls upon it and absorbs very little ; the
metal will therefore be very slightly heated. On the other
hand, a dull black surface absorbs practically all the radiation
which falls upon it, and the heating effect is correspondingly
greater.1 This may be illustrated by the following experi ment
(Fig. 99) :-

1 Cp. footnote on page 211.

RADIATION

215

To galvanometer

FIG. 99.

Two large tinned iron discs are mounted on stands to face
each other. The face of one disc is polished and that of the
other is coated with lampblack. On the back of each disc,
in the centre, is soldered a bar of bismuth. Each piece of
bismuth is connected by a wire to one terminal of a galvano-
meter and the circuit is completed by a wire connecting the
two discs. There are thus in the circuit two bismuth-iron
junctions ; if either junction is heated, a current tends to
flow round the circuit. But whereas the junction at A when
heated tends to make the current flow in the direction of the
arrow, the other junction when heated tends to make it flow in
the other direction ; l thus, if both junctions are equally heated
no current will flow. A hot copper ball is placed on a
stand midway between the discs, and it will be noticed that
the spot of light at once begins to move on the galvanometer
scale, showing that one junction is hotter than the other.
If, however, the hot ball is moved nearer to the polished disc,
the spot of light moves back and a position can be found,
much nearer the polished plate than the black plate where
there is no galvanometer deflection.

Thus it appears that when the two discs are receiving
similar amounts of radiation, the black surface absorbs a
greater proportion than the bright and so becomes hotter ;

1 With a bismuth-iron junction the direction of the current is from
bismuth to iron across the hot junction.

216

HEAT

it is not until the ball is brought nearer to the bright plate,
so that the intensity of the radiation upon it is much greater
than that falling on the black, that the heating effects become
equal. By coating the discs with different substances, the
effects of the surfaces may be compared.

It is found that there is a connection between the absorbing
power of a surface and its emissive power ; surfaces which
are good absorbers are good radiators also, and vice versa ;
moreover, accurate experiments, such as those of Melloni,
show that the emissive powers are proportional to the absorb-
ing powers, i.e. if one surface absorbs, say, 50 per cent, of
the incident radiation and another 25 per cent., .then the

surface will radiate twice
as much as the other
under the same condi-
tions.1

A well-known piece of
apparatus to illustrate
this point is represented
in Fig. 100. P and
T are two similar air-
tight tins, connected to-
gether by a glass U-tube
in which is contained
some coloured liquid.
The whole forms a differ-
ential thermometer ; if
P is heated the air in
IG' ' it expands, driving the

liquid to the right, while if T is heated the liquid moves

--s

--------

1 This statement requires a certain amount of qualification. If the
surfaces absorbed respectively 50 per cent, and 25 per cent, of the
radiation from a Leslie cube at 100° C. they would not necessarily
absorb the same fractions of the radiation from a copper ball at 300° C.
nor would one fraction be twice the other. But the radiating powers
of the surfaces would also be different at 300° C. from what they would
be at 100° C. ; the radiating and absorbing powers would correspond to
one another at 300° C. and also at 100° C.

RADIATION 217

in the other direction. If both are heated to the same
temperature the liquid remains in its original position. The
surfaces of the ends of P and T which face inwards are treated
differently, e.g. one may be highly polished and the other
lampblacked, or one may be painted. RS is a cylindrical
tin, which can be filled with hot water, mounted midway
between P and T and arranged so that it can be rotated about
a vertical axis. The surfaces of the ends of RS are made
similar to those of P and T, but if, for example, P is polished
and T is black, then R. is black and S is polished. On pouring
hot water into RS, no movement of the liquid in the air
thermometer takes place ; the radiation from R is greater
than that from S, but the smaller absorbing power of P and
the greater absorbing power of T just compensate the effect.
If, however, RS is rotated so that the two blackened surfaces
are opposite, a considerable movement of the liquid takes
place.

At this stage it is convenient to consider for a moment the
action of the thermopile. When the radiation falls upon one
end most of it is absorbed and the absorbed radiation is con-
verted into heat. The temperature of the receiving end
therefore rises, and continues to do so until the rate of loss
of heat to the surroundings is equal to the rate of supply of
heat by absorption of the radiation, when the temperature
remains steady and the deflection of the galvanometer is
constant. The final value of the deflection is thus a measure
of the rate at which the thermopile is absorbing radiation,
but the time taken to reach this final value depends on the
mass of metal in the instrument and has nothing to do with
the time taken by the radiation in passing to it from the hot
body. It will have been noticed that the spot of light on
the galvanometer scale begins to move immediately a hot
object is brought near, and in the case of the more accurate
instruments, the radio-micrometer and the bolometer, in
each of which the absorbing parts are of extremely small
mass, the indications are practically instantaneous. As we
shall see later, there is every reason to believe that radiant

218 HEAT

heat travels with a velocity similar to that of light (186,000
miles per sec. in vacuo, and somewhat less in air or other
substances).

Diathermanency. — Hitherto we have dealt only with
objects which absorb any portion of the incident radiation
which is not reflected from the surface. Some substances,
however, will allow radiation to pass through them. A sub-
stance through which radiant heat can pass is spoken of as
diathermanous, while one through which radiant heat cannot
pass is said to be adiathermanous. These terms correspond
to the expressions transparent and opaque in the case of
light ; a substance through which light can pass is trans-
parent, while one through which radiant heat can pass is
diathermanous. It does not by any means follow that a
transparent substance is diathermanous (e.g. glass is very
adiathermanous), nor that an opaque substance is adiather-
manous.

Some of the more important points may be illustrated by
the following method : In front of a thermopile, and a few
centimetres away from it, is placed a metallic screen with a
hole in its centre ; the hole should be considerably smaller
than the cone of the thermopile so that all radiation coming
through is collected by the cone.

A hot object, such as the hot copper ball, is placed at such
a distance from the opposite face of the screen as to give a
convenient deflection on the galvanometer scale. Plates of
different materials, of as nearly as possible the same thick-
ness, are placed in succession between the screen and the
thermopile so as to cover the opening but not touching the
screen. If a glass plate is interposed between screen and
thermopile the spot of light on the galvanometer scale will
move back nearly to zero ; the effect is even more marked
if the source of radiation is the blackened face of a Leslie
cube filled with boiling water. Glass is thus very adia-
thermanous. ^ A plate of alum will also be found to be
very adiathermanous. On the other hand, with a plate of

RADIATION 219

rock salt,1 the deflection is not much less than with the
open hole, showing that rock salt is extremely diather-
manous.

Another very diathermanous substance is a solution of
iodine in carbon di-sulphide, a solution which is opaque to
light. If a very thin- walled flask be placed empty between
the screen and the thermopile and the deflection of the
galvanometer noted (if the glass is very thin a reasonable
deflection will be obtained), and the flask be then filled with
the solution of iodine the deflection will be very little
diminished.

Water, on the other hand, is very adiathermanous, but the
presence of substances in solution appears to increase the
diathermanency ; even a solution of alum is more diather-
manous than pure water although the idea is still sometimes
met with that such a solution is one of the best adiather-
manous media.

We have just seen that glass is very adiathermanous ; on
the other hand, the action of a convex lens as a " burning
glass " is a well-known phenomenon. If a sharp image of
the sun is focussed, by means of a convex lens, on to a piece
of paper, the latter will be charred and may even burst into
flame. There is no question about the development of heat ;
if the hand be substituted for the piece of paper, the fact
is at once apparent. We can make the behaviour of the glass
more obvious by a further experiment with the apparatus of
the preceding paragraph. Place the hot copper ball at such
a distance from the screen as to give a suitable deflection on
the galvanometer scale, and observe that if a glass lens be
placed between the aperture of the screen and the thermopile,
the spot of light goes back nearly to zero. Remove the lens,

1 The surface oi rock salt tends to become pitted when exposed to
moist air owing to the hygroscopic nature of the salt ; for this reason
such plates are often supplied sealed between two glass plates. It
will be obvious that as glass is very adiathermanous, these plates must
be removed in the abvoe experiment if the effect of the rook salt is to
be seen.

220 HEAT

and substitute for the copper ball an arc lamp,1 placed at
such a distance from the screen as to give about the same
deflection on the galvanometer scale as was obtained from
the ball.

If now a slab of glass several centimetres thick be placed
between the screen and the thermopile, it will be noticed
that the deflection of the galvanometer is very little less than
without the glass. But if in addition a flask of iodine in
carbon di-sulphide be placed between the glass and the
thermopile, the spot of light will go back nearly to zero.
Thus in the case of the arc lamp we are getting results which
are almost exactly the converse of those with the copper ball-

Now the experiments with the copper ball showed that
radiant heat passed through the iodine solution, but not
with any readiness through glass ; on the other hand, we can
see that light passes through glass but not through the iodine
solution. It looks then as if light affects a thermopile in the
same manner as radiant heat, and in fact all experiments
show that this is so.2 For this reason the term " radiation "
is used in a general sense to include both light and what we
have called radiant heat ; when radiation of any sort falls
upon a surface some or all of it may be absorbed and that
part which is absorbed is in general8 converted to heat.

At this stage it will be convenient to consider some pro-
perties of radiation ; in the course of this consideration two
important points will emerge, i.e. that just as light may be
of different colours, such as red or blue, so radiant heat may
have different characteristics ; and further, radiant heat and
light are not two sharply distinguished kinds of radiation,
but the one merges into the other so that the boundary
between them is not easy to define, and, indeed, depends to
some extent on the observer. Anything like a full eonsidera-

1 The arc lamp will probably belong to an optical lantern, and it is
convenient to keep it inside the lantern body (to avoid an intolerable
glare in the room) but the condenser and all lenses must be removed.

* Cp. footnote, page 225.

3 An exception occurs in the case of fluorescence, where some part of
the absorbed radiation is converted to radiation of a different type.

RADIATION

221

tion involves the whole subject of spectroscopy and the
following treatment is only to be regarded as the barest
outline. We shall first deal with some properties of light and
then show how these properties extend to radiation in general.
If a narrow beam of light falls upon a prism of glass in a
direction such as AB (Fig. 101), it will suffer refraction at the

Fio. 101.

two faces ; that is to say, it will be bent (towards the thick
part of the prism) and come out in a direction making a con-
siderable angle with AB. The angle will depend on the
particular prism used, but it will also depend on the colour
of the light. A convenient experimental arrangement for
illustrating this point is shown in Fig. 102. A metal screen,
with a narrow vertical slit A, is arranged as shown, and

Fio. 102.

222 HEAT

behind it sources of light of different colours will be placed
in succession.

B is a convex lens adjusted so that if no prism were there
it would form a real image of the slit at about the position E.
When the prism C is in position, the light will be bent and
the image of the slit can be received on a screen at D. The
arrangement for getting lights of different pure colours is
shown in the diagram. F is a silica T tube ; in the side tube is
fixed a narrower tube connected to the gas supply and fur-
nished with a small hole at H. The gas issues from this hole
and is lighted at the top of the vertical branch of the T,
forming a sort of bunsen burner. G is a small flask containing
a solution of a salt, to which is added some hydrochloric acid
and a few pieces of magnesium ribbon. The magnesium
dissolves with a brisk evolution of hydrogen, causing a spray
of the salt solution to pass up to the flame and colour it. The
colour depends on the salt in solution.1

If a flask containing a solution of lithium chloride be placed
under the T tube, a pure red colour is obtained and the red
image of the slit will be seen on the screen in some such
position as shown at R. If this position be marked, and
(without moving any other part of the apparatus) a flask
with a solution of common salt be substituted for the lithium
chloride, a yellow light is obtained, but the yellow image of
the slit is not in the same position on the screen ; it will be
found in some such position as Y, showing that the yellow
light is bent through a larger angle than the red.

If now a flask containing both salt and lithium chloride be
substituted a yellow and red image of the slit will be formed
in the positions Y and R ; the prism sorts out the mixed
colour of the flame into the two constituents. Blue light is
bent even more than yellow ; it is not, however, so easy to

1 This arrangement, very convenient for spectroscopic work, has
been used most successfully by Prof. T. R. Merton. It is described in
Home detail in a lecture given by him of which an abstract appears in
The School Science Review, February, 1921. John Murray. The
dimensions of the apparatus are important.

RADIATION 223

get a salt which gives a blue light intense enough to show
up well on the screen. But if a Geissler discharge tube con-
taining hydrogen be substituted for the flame, the mixed
colour will be sorted out by the prism into a red image (not
quite in the same position as the lithium red, which is of a
slightly different shade), a greenish blue image beyond the
position of the sodium yellow, and a blue image beyond that.
There is a violet image beyond that, but it is not very easy
to see and may be missed. Thus light of each colour is bent
through a definite angle, so that each colour gives rise to an
image in a definite position.1

If now we substitute for the coloured flame a source of
white light such as an arc lamp, we do not get a single white
image of the slit on D, but a band of colour, one end of which
is red shading off gradually to orange, the orange to yellow,
yellow to green, green to blue and blue to violet, the latter
forming the other end of the band. Such a band of colour
is known as a continuous spectrum and represents an in-
definite number of overlapping images of the slit. It appears
then that white light is complex and the prism is able to sort
out from it all these different colours.2 Each kind of light
bends through its own particular angle and gives its own
definite image on D, and the position of the image identifies
the particular kind of light.

It has already been mentioned that light affects a thermo-
pile or similar instrument, and we can make use of this effect
to learn a good deal more about the spectrum. An ordinary
thermopile is not suitable for this purpose, since it would
cover too large a part of the spectrum at once. A more
suitable instrument is a Rubens linear thermopile, with the
junctions arranged in a single vertical row ; a bolometer or a
radio micrometer would generally be used in accurate work.

By making a slit in the screen D, not quite so wide as the

1 Provided that we keep to the same prism throughout and do not
alter the direction in which the incident light meets the prism face.

2 Strictly speaking the matter is not so simple, and modern work,
begun by Gouy as far back as 1886, shows that th*» action is not a mere
sorting out of colours already there.

224 HEAT

height of the spectrum but extending well beyond it at each
end, and placing the linear thermopile behind it, we can
examine the heating effect at each part. It will be found
that when the thermopile is in the violet part the heating
effect is small, but as it is moved towards the red end the
effect is increased. As the instrument is moved out beyond
the red end there is no sudden change ; a heating effect is
found in a region extending well beyond the visible spectrum.
As the movement is continued the heating effect gradually
diminishes and finally becomes incapable of detection. The
radiation from the arc lamp is thus sorted out by the prism
into a long spectrum, part of which is visible and part
invisible.

The invisible spectrum must consist of parts of different
qualities corresponding to the different coloured lights, for
the parts of the invisible radiation are bent through different
angles.

Now the study of the properties of light (chiefly the
phenomena of interference and diffraction) led to the con-
clusion that light is some form of wave motion. As, how-
ever, it travels across a vacuum, it cannot be a wave motion
in a material medium such as is met with in the case of sound.
But a wave motion implies some medium for its transmission,
and so there was developed the idea of a medium filling all
space. This medium is spoken of as the aether, and light is
regarded as some form of wave motion in the aether.1 The
different coloured lights are regarded as being of different
wave lengths (the distance from one wave to the correspond-
ing point on the next). These wave lengths have been
measured by means of diffraction gratings and later by more
accurate methods, and it is found that the wave length of red
light at one end of the spectrum is nearly twice as great as
that of violet at the other, the values being about -00077 mm.

1 The precise meaning to be attached to the term aether is a question
far beyond the scope of this book. The wave motion is almost certainly
some form of electro -magnetic disturbance, but the precise method of
its propagation is a matter difficult to decide.

RADIATION 236

and 0004 mm. respectively. Intermediate colours have wave
lengths somewhere between these amounts. Thus white
light may be regarded as containing radiation of all wave
lengths between these limits and the prism sorts them out
and sends them in different directions, like a policeman
directing traffic.1 The thermopile indicates that the spectrum
does not end at the red, but extends without interruption for
some distance beyond ; in this case there is no distinction
between visible and invisible radiation.2

This leads to the conclusion that the invisible radiation is
similar to light, i.e. some form of wave motion in the aether,
but of wave length ranging from that of the extreme red
light to greater, values. Measurements made by Langley
(Phil. Mag., 1884 and 1886) with a diffraction grating used
in the same way as in the case of light, determined some of
these longer wave lengths and many others have been made
subsequently. The radiation from a source such as an arc
lamp must be regarded as made up of constituents having
every wave length between certain limits. All of this radia-

1 The really essential characteristic of any light is not so much the
wavelength as the frequency, i.e. the number of waves passing a given
point per second. So long as the light moves in one medium it does
not matter whether we deal with frequency or wave length, but if the
light passes from one medium «o another the wavelength changes while
the frequency does not. r

Since the frequency is the number of waves passing a given point
per second it follows that frequency X wave length = distance the light
travels in unit time, i.e. the velocity.

2 At one time the view prevailed that light did not itself produce
heating effect, but that this was due to a separate kind of radiation.
When it was found that after sorting out the radiation with a prism a
heating effect was still observable in the visible part, it was thought
that possibly there was some portion of invisible radiation which could
not be separated from the light by the prism. To settle the point
M. Jarnin (Comptes Rendus, 1850) allowed a beam of the light from
one portion of the spectrum to pass through various absorbing media,
and made measurements : (a) by a photometer of the diminution of the
light, (6) by thermopile of the diminution of the heating effect. He
found that in all cases the percentage of diminution of the heating
effect was equal to that of the light and concluded therefore that heat-
ing and light were merely properties of the same radiation. An accoun t
of this work is given in the admirably clear " Etudes des Radiation "
in M. Jamin's Cows de Phi/sique, 1891. torn. 111.

226 HEAT

tion affects a thermopile, but some of its properties depend
on the particular wave length, and only those parts of it are
visible of which the wave length is less than about -0008 mm. ;
the precise value of this limit is found to vary slightly for
different people. All this invisible radiation of wave length
greater than about -0008 mm. is spoken of as infra red radia-
tion, a much more satisfactory expression than radiant heat.

Transmission of Radiation through Different Media.
— As soon as it is realized that radiation may have different
characteristics according to its wave length, many of the
apparent anomalies in connection with diathermanency and
adiathermanency are accounted for. Many substances will
allow radiation of certain wave lengths to pass through them,
but absorb strongly radiation of all other wave lengths. Glass
readily transmits radiation of all wave lengths between about
•0004 mm. and -002 mm., that is to say, all the visible radiation
and a considerable range of the infra red, but absorbs strongly
anything of longer wave length. Alum transmits the visible
radiation, but very little else.

Iodine in carbon di-sulphide transmits the long wave radi-
ation but absorbs the visible. Rock salt appears to transmit
all radiation readily,1 and so also does sylvine (potassium
chloride).

The Connection between the Temperature of a Body
and the Characteristics of the Radiation it Emits. — Not
only does the total amount of radiation from a hot object
depend on the temperature, but the character of the radiation
varies also. For example, suppose we have a copper ball at
a temperature of about 100° C. It is obviously not luminous,
but the radiation can be sorted out into a spectrum (but it
will be necessary to use a lens and prism of rock salt, since
glass absorbs the long wave radiation) which can be explored
with a linear thermopile or a bolometer.

1 It absorbs strongly the particular type of radiation given off by
another heated piece of rock salt ; this is a particular instance of a
well-known phenomenon met with in apectroscopy.

RADIATION 227

The spectrum will not be very long, and its position will
indicate that it is composed of relatively long wave-length
radiation. If the temperature of the ball be raised, explora-
tion with the thermopile will show that the spectrum has
lengthened out in the direction of the visible, showing that
the radiation now contains some shorter wave-length con-
stituents.

If the temperature is raised still further the spectrum
lengthens out, and at something like 700° C. it just reaches
the position of the extreme red and the ball begins to appear
a dull red. At higher temperatures other components of the
visible spectrum appear, and at something like 1,300° C. all
the colours are present and the ball is " white hot."

With a source of radiation at still higher temperature, such
as an arc lamp, it is possible to detect an extension of the
spectrum beyond the violet. As a rule only the most sensitive
of radiometers will detect the heating effect of this very short
wave-length radiation, and of course it is not visible. It can,
however, be detected by its action on a photographic plate
or by allowing it to fall on a fluorescent screen such as is used
for X-ray purposes. This radiation is spoken of as ultra
violet radiation.

Glass is very opaque to ultra violet radiation, but rock salt
and sylvin transmit it readily, as does quartz.

There is one point in connection with the radiation spectrum
of a hot body which is of great importance in practical pro-
blems of illumination.

If the radiation from such a body be spread out into a
jpectrum by means of a prism (of rock salt or fluor spar) and
the spectrum be explored with a bolometer or linear thermo-
pile, the reading of this instrument will be a maximum in one
position and will gradually diminish to zero as it is moved
away in either direction. If the readings of the thermopile
are plotted for the different positions the curve would be
something like Fig. 103, ABC. If now the temperature of
the body be raised and the spectrum again explored, the
result will be as indicated by DEF. In this curve it will be

228

HEAT

seen that the spectrum extends further into the shorter wave-
length region, and that the position of maximum energy
corresponds to a shorter wave length than in the case of ABC.
Notice, however, that the whole of DEF is above ABC, i.e.
the body does not cease to give out the long wave-length
radiation ; in fact it gives out rather more, but there is a
more than proportionate increase in the amount of short
wave radiation. The higher the temperature of the body the

Ultra rwlet Infra, Red

Wavelength.

FIG. 103.

further up the spectrum does the position of maximum
energy appear. In the case of the sun, whose effective tem-
perature (see page 253) is probably somewhat under 6,000° C.,
the position of maximum energy is in the yellow of the visible
spectrum ; a body at 1,500° C. would give the position of
maximum energy not in the visible spectrum at all but well
in the infra red.

Now the total energy radiated will be proportional to the
area under the whole curve, while the amount of light (the
part of the radiation which is visible) will be proportional to
the area under the curve between the values of the wave-
length which correspond to the upper and lower limits of the
visible spectrum. If the position of maximum energy is well
in the infra red it is clear that very little of the total radiation
is light, while if the position of maximum energy is some-
where near the middle of the visible spectrum, a large pro-

RADIATION 229

portion of the whole radiation will be light ; this is in fact
the condition in the case of the sun's radiation. Thus the
higher the temperature of a body the greater will be the
proportion of its total radiation which is in the form of light
(unless indeed it were hotter than the sun, a condition not
likely to be realized).

The arrangement in an ordinary incandescent electric
lamp is to send a current through a thin wire or filament
enclosed in a glass bulb. The electrical energy is converted
to heat and the wire becomes white hot ; its temperature
depends on the length, material and thickness of the wire.
The amount of electrical energy used will be proportional to
the total emission of heat from the wire, and as we have just
seen the proportion of this energy which appears as light
increases as the temperature is raised. The lamp will there-
fore be more efficient the higher the temperature at which it
is arranged that the filament shall be. The difficulty is to
find a material which is able to stand a very high temperature
without damage. In the early lamps a carbon filament was
used. Such lamps normally take about 4 watts per candle-
power. They can be made much more efficient than that by
running at a higher temperature, but the filament rapidly
disintegrates and soon breaks down. The next stage was to
use a filament of tungsten ; in this case the metal filament
can be run at a much higher temperature and yet last a long
time. Such lamps only use a little over 1 watt per candle-
power. If, however, they are made to run at a still higher
temperature they soon disintegrate. Disintegration can be
considerably diminished by having the bulb filled with an
inert gas, in which case the filament can be run at a still
higher temperature. At first, however, the conduction and
convection losses due to the gas more than compensated the
gain of the higher temperature ; this trouble was overcome
by a special arrangement of the shape of the filament which
in effect diminishes the area of surface available for conduction
to the gas as compared to the radiating surface. The fila-
ment is wound in a spiral of extremely small pitch. The

230

HEAT

modern gas-filled lamp (using argon as the inert gas) only
uses a trifle over J watt per candle-power.

Thus it will be seen that the problem of lamp design was to
get a filament which could be run at a very high temperature
and yet last a reasonable time. An ordinary carbon lamp
can be made to give 1 candle-power per half-watt by over-
running it (i.e. putting it on a circuit where the voltage is
much greater than that for which it is designed), but in that
case the filament usually lasts only a few minutes.

Some Practical Considerations.

The properties of radiation, which have just been con-
sidered, have an important bearing on many practical
problems, of which one or two may be mentioned.

Heat Insulation.— The Thermos Flask.— The object of
the thermos flask is to prevent as far as possible the contents
from losing heat to the surroundings or gain-
ing heat from them ; the original apparatus,
known as a Dewar vacuum vessel, was de-
signed chiefly for the latter purpose, and was
used for containing liquid air and other lique-
fied gases. The vessel (Fig. 104) is double
walled and the space between the walls is ex-
hausted to a very high vacuum. This forms
the best possible obstacle to the transference of
heat by conduction or convection across the
space. Radiation, however, travels more readily
across a vacuum than when there is a material
medium ; advantage is therefore taken of the
fact that a highly polished metallic surface is a
very bad radiator. The outside of the inner vessel and the
inside of the outer are therefore silvered and given as bright
a surface as possible ; radiation from the surface of the
inner vessel is thus reduced to a minimum, and very little
of this small amount of radiation is absorbed by the outer
vessel but most of it is reflected. In the same way radiation
from the outer vessel to the inner is equally diminished.

FIG. 104.

RADIATION 231

The mouth of the flask is usually closed by a cork, so that
there is a slight conduction leak here and also across the top
through the glass ; the latter is therefore made as thin as
possible consistent with strength. These leaks are small,
and the arrangement forms the most efficient heat insulator
available. The flask being rather fragile, is surrounded by
an outer metallic vessel, and a spring at the base and suit-
able soft packing are provided to protect it from shocks.

Radiation from the Earth. Screening Action of
Clouds. — It is a matter of common observation that it is
colder on a clear night than when the sky is heavily clouded.
In both cases the earth is radiating ; on the clear night the
radiation can pass out freely, but when clouds are present
they act as a screen.

Some of the radiation which falls upon the clouds is irregu-
larly reflected back towards the earth, and the remainder is
practically all absorbed since water is very adiathermanous
to long wave radiation. The absorbed radiation raises the
temperature of the cloud, which therefore radiates on its
own account.1 Some of this radiation is sent towards the
earth. Thus the earth receives radiation and so the net losa
of heat is less rapid than when the radiation passes away
without interruption.

To a lesser extent the presence of a large amount of water
vapour in the air will have an effect in reducing the rate of
loss of heat ; water vapour is distinctly adiathermanous, and
so absorption of radiation takes place and subsequent radia-
tion back ; in a very dry climate therefore the nights tend
to be colder than in a moist one.

Selective Absorption of Glass. — Two examples of this
may be quoted. Glass is sometimes used in fire-screens ; it
allows the light to pass through, and also a little of the infra
red radiation. It absorbs strongly all the infra red radiation
below a certain wave length, and since with an ordinary fire
the greater part of the total radiant energy is contained in

1 See page 233* " Theory of Exchanges."

232 HEAT

constituents of long wave length, the screen will cut out a
large part of the heating effect but allow the fire to be seen.

Another example is seen in the action of an ordinary
greenhouse, or any glass-roofed building. When sunlight
falls upon the glass the visible portion of the radiation and
the shorter wave length part of the infra red is transmitted,
while the longer wave length portion of the infrared is absorbed
and the glass is thereby heated. The sun is at a very high
temperature (something like 6,000° C.), and consequently a
considerable portion of the total energy radiated is of short
enough wave length to be transmitted.

The transmitted radiation is absorbed by the floor and
objects inside the greenhouse, which become heated. When,
however, they radiate, the radiation is of long wave length
(since they are not nearly hot enough to begin to be luminous),
and is not transmitted through the glass. The glass thus
absorbs most of this radiation and radiates back to the
interior, just as in the case of the cloud in the last paragraph.
The effect, however, is much greater because the glass is
already absorbing radiation from the sunlight as well as the
radiation from the interior. It thus radiates more than it
receives from the interior, and the temperature of the interior
is raised. The heating of the interior is thus due to the
selective absorption of glass which lets through a large part
of the solar radiation but will not let through the longer
wave-length radiation from the objects inside the greenhouse.

CHAPTER XV
LAWS OF RADIATION

Prevost's Theory of Exchanges. — From the foregoing
account of the properties of radiation, it might be assumed, in
the absence of any statement to the contrary, that a body only
begins to radiate when it is at a higher temperature than its
surroundings ; so that if a hot body is placed in an enclosure it
will radiate to the surroundings until it has reached the tem-
perature of the enclosure and then stop. We shall find, however,
that if we adopt this point of view it will be difficult to account
for many observed phenomena ; a much more helpful way of
regarding the matter is to consider that radiation is going on all
the time, and that when the hot body reaches the temperature of
the enclosure it does not stop radiating, but an equilibrium is
reached in which the amount of radiation given out by the body
is equal to the amount it absorbs from the surroundings. This
theory was proposed by Prevost to account for results of the type
indicated in the next paragraph ; it was developed quantitatively
by Kirchoff and Balf our Stewart and has been of great importance
in the theory of radiation.

If a lump of ice be brought near a thermopile, the galvanometer
immediately indicates a cooling effect as if the ice were emitting
a cold radiation. A more remarkable effect is observed if two
large concave mirrors are placed facing one another and a ther-
mopile is placed at the principal focus of one of them. If a piece
of ice be placed at the principal focus of the other a very marked
cooling effect is indicated by the galvanometer, but if the ice is
shifted from this position the cooling effect is much less. The
effects are precisely analogous to the heating effect observed if
a hot object be substituted for the ice.

Let Fig. 105 represent an enclosure, the walls of which are
maintained at some particular temperature by external means.
Let a body A at a different temperature be placed in the enclosure.

233

234 HEAT

Experience tells us that A will ultimately come to the same
temperature as the walls of the enclosure ; this might be ex -
pressed that A loses heat by radiation to the walls or gains heat
by radiation from them until it attains their temperature.1

According to the Theory of Exchanges, however, the walls are
radiating to A and A is radiating to the walls ; the temperature
of A will rise or fall according to whether the body absorbs more
radiation than it emits or vice versa. When these two amounts
become equal the temperature remains steady, but the radiation
is regarded as still going on, so that the enclosure is crossed by
streams of equal and opposite radiation. If other bodies, B, C,
are placed in the enclosure, each of these attains the temperature
of the walls, whether the others are present or not.

Since the presence of these other bodies does not affect the
temperature attained by A wherever
it may be placed in the enclosure,
it follows that they do not produce
any alteration in the radiation streams
in the space. This means that if they
absorb any part of the radiation which
falls upon them they must also radiate
FIG. 105. an equal amount. This involves the

condition, which we have already seen

has been found by experiment, that a substance which is a good
radiator is a good absorber.

For suppose A is a piece of metal with a dull black sur-
face, B is a piece of metal with a polished surface, and C
a piece of rock salt. A absorbs most of the radiation fall-
ing on it from all sides, but radiates a large amount in all
directions also, so that we get our opposing streams of radia-
tion. B reflects most of the incident radiation and absorbs very
little. The reflected radiation has only to be supplemented by
the small amount which a body with polished surfaces can radiate
to produce the opposing streams as before. C transmits nearly
all the radiation incident from all sides ; the radiation from the
walls on one side passes through it to the other and we get our
opposing streams as before. Very little radiation from C is

1 If there is any material medium in the enclosure, conduction or
convection may assist in the process, but the final result would be the
same if the space were vacuous.

LAWS OF RADIATION 235

required to restore the small amount absorbed by it from the
general stream.

It follows also that not only does each body in the enclosure
emit a quantity of radiation equal to that which it absorbs, but
it emits radiation of the same quality also. For if it did not the
quality of the radiation streams would be altered by the presence
of such a body ; the proportion of radiation of some particular
wave length would be increased.

If then we had some substance which absorbs radiation of this
particular wave length strongly and other radiation very little it
would absorb more energy from the altered radiation stream than
it would from the original. But what it radiates would depend
only on its own nature and temperature, and so it would rise to
a higher temperature than under the original conditions. This
however is contrary to experience ; all bodies in a- constant tem-
perature enclosure ultimately reach that temperature whether
other bodies are there or not, so that the presence of these bodies
does not alter the character of the radiation stream.

This connection between the quality of the radiation which a
substance can absorb and that which it radiates is a well-known
experimental fact, to which allusion has already been made (see
footnote, page 226). Rock salt, which is very diathermanous to
most kinds of radiation, absorbs strongly the radiation from a
heated piece of rock salt. A piece of red glass, i.e. glass which
transmits red light but absorbs strongly light of all other colours,
will if heated to a high enough temperature give out light of the
colours which it absorbs and appears bluish. A dull black
surface which absorbs radiation of all wave lengths almost equally
well (but no perfectly black surface has been produced of which
this is strictly true) can radiate all wave lengths if raised to the
required temperature, and so on.

Full Radiation. — Since the presence of various bodies in
the enclosure does not alter the character of the radiation streams
when temperature equilibrium has been attained, it follows that
these streams must contain radiation of all wave lengths which
a body at that temperature could emit ; moreover the energy
density (amount of energy per unit volume) for each constituent
must be the maximum for that temperature. For if one of
the bodies were " perfectly black " (i.e. absorbed 100 per cent, of
any radiation incident upon it, whatever the wave length) the
radiation from it would correspond to the general stream j the

236 HEAT

radiation from all other bodies would merely have to make up
the deficit in the general stream caused by their partial absorp-
tion. For this reason the radiation stream inside a constant
temperature enclosure is spoken of as the " full " or " complete "
radiation corresponding to that temperature ; and if we have
such a constant temperature enclosure with a small aperture
through which the radiation can escape the radiation from it will
correspond to the radiation from a perfect black body whose
surface coincided with the aperture.1 Since the objects inside
a constant temperature enclosure attain the same temperature
whatever the nature of the walls, it follows that the radiation
stream inside is independent of the material of the walls, a fact
which is borne out by experiment. The full radiation corre-
sponding to any temperature is therefore something which is
independent of the properties of any substance.

The Theory of Exchanges. — The Theory of Exchanges
(which as we have seen starts with the idea that a body at any

temperature is always giving out
radiation, depending only on the
temperature and nature of the
body, and that when its condition
remains unchanged it is absorbing
radiation at a rate equal to that
at which it is radiating) is very
FIG. 106. helpful in simplifying our ideas and

enabling us to predict what will

happen in any given conditions. Let us apply it to the case
mentioned on page 233.

The thermometer placed at A (Fig. 106) at the principal focus
of one mirror is radiating in all directions and is receiving radia-
tion from its surroundings (the walls of the room and other
objects). Let us suppose it has taken up a constant tempera -

1 In the further development of the Theory of Radiation difficulties
arose which caused some doubts as to the validity of the assumption
that full radiation and black body radiation are identical. These
difficulties can be overcome, e.g. in Planck's theory, but sortie writers
prefer to define full radiation as the radiation in an impervious enclosure
and avoid committing themselves as to its relation to black body
radiation. The discussion of these points is a good deal beyond the
scope of this book ; a comparatively simple account is given in N. R.
Campbell's Modern Electrical Theory.

LAWS OF RADIATION 237

tore. The rate at which it absorbs radiation must then be equal
to that at which it emits it. Those portions of the radiation from
surrounding objects which happen to pass through B and fall on
the mirror GH (B is the principal focus of GH) will suffer reflec-
tion at the two mirrors and fall on A ; 1 the mirrors themselves
as polished surfaces will radiate very little.

Thus nearly all the radiation stream which reaches A from
directions between DA and CA must have passed through B.
Now let a piece of ice be placed at B. A good deal of the radia-
tion coming up to B will be absorbed by the ice which is adia-
thermanous for long wave radiation and so will not reach A.
The ice will radiate, but being at a low temperature will not
radiate so much the objects in the room (assuming that the room
is above 0° C.), and so the radiation reaching A by reflection at
the two mirrors will be less than before. The thermometer goes
on radiating, but is now receiving less than it did before, and so
the absorbed radiation no longer balances that emitted and the
temperature falls. Putting the ice at B enables it to intercept
a larger part of the radiation streams travelling to A than it
would do anywhere else (except of course if the ice were so close
to A as to subtend a bigger angle than DAC), and so its effect is
most marked in this position.

Rate of Cooling of a Hot Body. — When a body is at a
higher temperature than its surroundings it will cool, and the
rate of loss of heat depends on the nature of the body and on the
temperature. For many purposes it is important to know what
the relation is between temperature and rate of loss of heat.
The earliest attempt to express this relation is due to Newton, and
is known as Newton's Law of Cooling. This Law states that the
rate of cooling of a body is proportional to the excess of its
temperature above that of its surroundings. It is sometimes
forgotten, however, in speaking of this law that Newton was
referring to particular conditions ; he was concerned with bodies
placed " in a cold place where the wind blows," e.g. a draughty
cellar, so that the loss of heat was mainly due to conduction and
convection. Even in fairly still air, such as that of a room, the
law is approximately true, if the temperature excess is not large,
so that the radiation loss is a small fraction of the total loss. It

1 A small fraction of it will be absorbed by the mirrors since the
surface is not a perfect reflector.

238 HEAT

is, for example, sufficiently near the truth to be used for making
a correction for cooling in calorimetry if the temperature of the
calorimeter does not rise more than 6 or 7 degrees above the room
temperature. It is not .true if the temperature excess is largo,
and it is certainly not true when the cooling is due to radiation
only and conduction and convection are eliminated.

Radiation Laws. — D along and Petit * made a long series of
experiments on the rate of cooling of large mercury-in-glass
thermometers placed in an enclosure kept at constant tempera-
ture. Experiments were made both when the enclosure con-
tained gas and when it was exhausted to the best vacuum they
could obtain. By these methods the conduction effect was
separated as far as possible from the radiation.2 They found
that the rate of cooling was not proportional to the excess of
temperature, but increased more rapidly as the temperature was
raised. They found that the actual value of the temperature of
the thermometer and enclosure mattered ; the rate of cooling
when the thermometer was at say 60° C. and the enclosure at
40° C. was greater than when the thermometer was at 50° C. and
the enclosure at 30° C. They found that there was a simple
relation between the rate of cooling for a given excess of temper-
ature of the thermometer over that of the enclosure and the
temperature of the enclosure. From this relation a formula was
deduced which was regarded as representing the radiation Law.
Later experiments have shown that this formula does not satis-
factorily express the relation between radiation and temperature,
and the work of Dulong and Petit is of interest chiefly as the first
really systematic investigation. After the time of Dulong and
Petit a number of experiments were made on the rate of cooling
of hot bodies and also the rate of emission of energy from a wire
heated by having a current of electricity passed through it ; in
this case the rate of loss of energy is obtained from the strength
of the current and the potential difference between its ends.

Stefan's Law — In 1879 Stefan suggested that the total
radiation from a body was proportional to the fourth power of its
absolute temperature. He was led to this idea by considering

1 Ann. de Chimie et de Physique, 2e Ser., torn. VII, 1817.

2 Strictly speaking, the effect of the gas is not confined to conduction
and convection. The rate of radiation is somewhat influenced by the
medium. Cp. B<alfour Stewart's Heat or Poynting and Thomson's
Heat.

LAWS OF RADIATION 239

the results of an experiment of Tyndall's on the rate of emission
from a platinum wire. He also examined the results of Dulong
and Petit from the point of view of this law and found that they
agreed with it very well ; better in fact than with the law given
in their paper. It is worth while considering precisely what is
meant by this fourth power law.

If we have a body of which the area of surface is A sq. cm.
and the temperature is 0° C., the energy radiated per second is

Acr(0 + 273)*,

where a is a constant depending on the nature of the surface.
But at the same time it is absorbing radiations from its surround-
ings and so the nett loss of energy is the difference between what
it radiates and what it absorbs. Suppose that it is placed in an
enclosure at 15° C. When the body is at 15° C. its temperature
remains constant. It must therefore absorb as much from the
enclosure as it radiates. But at 15° C. the energy it radiates per
second is

A<r(15 + 273)*

and this must also represent what it absorbs from the radiation
of the enclosure.

When it is at 0° C. the nett rate of loss of energy by radiation
must be *

A<r(0 + 273)«-A<r(15 + 273)*.

It is instructive to take a numerical example. Let us compare
the rate of loss of heat by radiation for a body when its temper-
atures are respectively 1,000° C. and 500° C. and the enclosure is
at 15° C.

In the first case the loss is A<r(1273)« - (288)*.
In the second case the loss is A<r(773)« - (288) «.

(1273)*— (288)4

The ratio of these is (773x4 _ (288)4' since A(T ^ the same
for each.

2625 X 10» - 6-88 X 10» 2618
This is equal to 357.2 x 10» - 6-88 x 10' = 350^3 = 7'45'

1 This assumes that the absorptive power of the body for the radi-
ation from the enclosure does not depend on the temperature of the
body. There is some ground for supposing this ; but as we shall see
later Stefan's Law is only strictly true for a perfect black body, and
such a body by definition absorbs 100 per cent, of all radiation which
falls on it whatever the temperature, in which case of course our con-
dition is satisfied.

240 HEAT

Thus doubling the temperature has increased the radiation
nearly 7^ times.

With Stefan's Law we get for the first time the relation be-
tween radiation and temperature on a satisfactory basis. Never-
theless the results of some observers are not in accordance with
this law, and it is important to realize in what the value of the
law consists and also the conditions in which it does not apply.

Boltzmann showed that the law could be deduced from con-
siderations of thermodynamics and the electro-magnetic theory
of light in the case of full radiation for an enclosure. A number
of experiments have been made to verify the law, of which the
work of Lummer and Pringsheim l is the classical example.
Using for the radiator an enclosure kept at a constant tempera-
ture and provided with a small aperture, they measured the
radiation with a surface bolometer at different distances and
found that the law was well obeyed over a very wide range. It
may therefore be regarded as established that the law holds for
full radiation, and therefore in the case of a body which approxi-
mates to a perfectly black body the law represents sufficiently
closely what happens. When, however, we are dealing with bodies
which do not approximate to black bodies considerable depar-
tures may arise, especially in the case of those substances which
show well-marked selective absorption.

Radiation Laws concerned with the Distribution of Energy
in the Spectrum. — It has already been stated (page 227) that
if the spectrum of a hot body were explored with a bolometer the
energy is a maximum for some particular wave length depending
on the temperature. Formulae have been deduced from theo-
retical reasoning for the distribution of energy in the case of full
radiation.

These formulse may be referred to here.

If Xm is the wave-length corresponding to the position of
maximum energy in the full radiation for a temperature 0°
(Absolute), then

\me = constant (Wien's Displacement Law).

The emissive power for the wave length of maximum energy is
proportional to the fifth power of the absolute temperature.

Lastly, a formula giving the emissive power for any particular

1 Ann. der Physik, Bd. LXIII. 1897. A good account is given in
Preston's Theory of Heat, Third Edition.

LAWS OF RADIATION 241

wave length has been deduced by Planck. These formulae only
apply to full radiation ; they do not apply in the case of a body
which is not perfectly black.

An elaborate series of experiments were made by Lummer and
Pringsheim to test the truth of these formulae, using a constant
temperature enclosure as the source of radiation. The results
showed in all cases a very good agreement between observed
values and those given by the theory. The discussion of these
formulae is beyond the scope of this book.

The chief point is to realize that they have been deduced and
that together with Stefan's formula they have been found to
represent the facts for full radiation. They are of importance in
the determination of temperatures by radiation as described in
the next chapter.

CHAPTER XVI

ON THE MEASUREMENT OF TEMPERATURE IN
INDUSTRIAL AND OTHER PROCESSES

Expansion -of -liquid Methods. — The ordinary mercury-
in-glass thermometer is an example of the use of expansion-
of-liquid method, but such an instrument is in many ways
unsuited for industrial processes. What is required is a
quick-acting instrument not likely to be damaged by moder-
ately rough usage, and in a number of cases it is desirable to
have the indicator at some distance away from the place
where the temperature is being measured.

A very useful form, suitable for temperatures up to 540° C.
(1,000° F.),is the steel-bulb mercury thermometer (Fig. 107).
A steel bulb is connected by flexible steel capillary tubing to
a flat steel tube coiled in a spiral and forming the Bourdon
tube of a dial gauge. The system of bulb, capillary tube and
Bourdon tube is completely filled with mercury under pres-
sure. When the bulb is heated the mercury expands and
this tends to uncoil the Bourdon tube and so moves a pointer
over the dial of the gauge. The latter is graduated to show
degrees, and by having a suitable length of capillary tube
the dial can be placed at any convenient place away from
the source of heat.

Such instruments are suitable for temperatures between
— 40° C. (the freezing-point of mercury) and 540° C. ; the
mercury, completely filling the system and being under pres-
sure, is not able to boil. Above 540° C. mercury begins to
attack the steel and incidentally the pressure developed at
this temperature is very large.

There is a slight error due to the uncertain temperature

242

INDUSTRIAL PYROMETRY

243

of the mercury in the capillary tube, and for this reason
75 feet is usually quoted as the maximum length of tubing
permissible. In the pattern produced by Messrs. Negretti
and Zambra, the error due to changes in temperature of the
capillary tube is overcome by an ingenious compensating
device. The capillary tube is made of
high-expansion material and has run-
ning throughout its length a wire of low-
expansion material (invar), slightly
smaller in diameter than the bore of the
tube. The wire is cut into lengths so
that it can slide in the tube when the
temperature changes. The mercury
occupies the annular space between the
tube and the wire ; by suitably adjust-
ing the diameter of the wire it can be
arranged that the expansion of the
tube just equals the expansion of the
small volume of mercury in the annular
space, and so the mercury in the tube
just fills its available space whatever
the temperature of the latter, and so
the indications on the gauge are not
appreciably affected by variations in
the temperature of the tube.

Vapour Pressure Thermometers.
— In these instruments advantage is

taken of the fact that the pressure of the vapour of any
substance in contact with the liquid (saturation pressure)
depends only on the temperature and not on the volume
of the container. In appearance the thermometer, which
was developed by J. B. Founder, is somewhat similar to the
steel-bulb thermometer mentioned in the preceding paragraph.
The bulb contains the liquid and its saturated vapour, and
the pressure is transmitted through the capillary tube to a
pressure gauge graduated to read degrees. It will be obvious
that changes in the volume of the bulb or capillary tube

FIG. 107.

244 HEAT

will not affect the reading of the pressure, and the gauge can
be at any distance. As the vapour pressure of a liquid in-
creases very rapidly with temperature, it is not satisfactory
to attempt to use the same instrument to cover a very wide
range of temperature ; separate thermometers are made in a
variety of ranges. Thus one instrument might be used for
a range --25° to + 25° C., another 180° C. to 340° C.,
another 500° to 850°. They can be used for temperatures
up to a value considerably above that of the steel-bulb
mercury thermometer. A type is also made to show clearly
small changes of temperature between two limits not widely
separated.

Electrical Pyrometers. — Electrical pyrometers may be
divided into two classes : resistance thermometers and thermo-
electric thermometers. The action of the former class is
based on the fact that the electrical resistance of a platinum
wire increases as its temperature is raised. The thermometer
consists essentially of a coil of fine platinum wire wound on
a frame of insulating material l and having its ends connected
by leads to terminals. The whole is protected from injury
by being enclosed in a tube of porcelain or quartz. The
instrument is connected to some form of Wheatstone bridge
which enables its resistance to be measured : the bridge is
calibrated so as to show the temperature directly. A change
in the temperature of the leads and connections would cause
a change in their resistance and hence an error in the resulting
observation ; thermometers for accurate work are made with
a second pair of dummy leads side by side with the leads of
the coil, and these are connected to the bridge by wires
similar to those connecting the real leads. The bridge is
arranged to measure the difference between the resistance of
the coil + its leads and that of the dummy leads. Thus,
any changes in resistance of the leads are automatically
compensated and the resistance of the coil only is measured.

1 The most suitable material appears to be mica, but some forms of
successful industrial instruments are woijnd on quartz.

INDUSTRIAL PYROMETRY 245

Of the special types of bridge for this work the two best
known are the Callendar Recorder and the Whipple Indi-
cator : the scale of the latter is calibrated to indicate single
degrees from — 10° C. to 1,200° C. For some types of in-
dustrial work the use of dummy leads can be dispensed with
and the indicator calibrated to read sufficiently accurately.
Platinum resistance thermometers can be used over a very
wide range of temperature ; they can be used for low tem-
peratures such as — 200 ° C. and can also be employed for
temperatures up to about 1,400° C. The upper limit of
temperature is fixed by the fact that no material for the
protecting tube has yet been found which is capable of
satisfactorily withstanding higher temperatures than about
1,400° C.

The two most important points in the design of these
instruments are efficient insulation and the avoidance of
mechanical strain in the coil ; the latter would cause a change
in resistance so that the instrument would be inconsistent,
i.e. would not always give the same indication for the same
temperature.

Thermo-electric Pyrometers. — These instruments, like
the thermopile, depend on the fact that if we have a circuit
of two different metals and one junction is heated, a current
of electricity tends to flow round the circuit. The electro-
motive force depends on the kind of metals used and on the
difference in temperature between the hot junction and the
cold. The apparatus consists of the " thermo-couple " (two
dissimilar metals joined at one end), which is connected to
an indicator which is some form of galvanometer calibrated
to read the temperature direct. The wires of the thermo-
couple are protected by a tube as in the case of the platinum
resistance thermometer.1

For temperatures up to 1,400° C. the most suitable materials

1 This is for general and industrial use. In the case of laboratory
use there are occasions when it is better to use a thermo-couple not
enclosed in a tube, and it may also be better to use a potentiometer
instead of the galvanometer.

246 HEAT

for the thermo-couple are platinum and an alloy consisting
of platinum with 10 per cent, of rhodium.

For temperatures up to about 600 or 700° C. other metals
can be used and in some respects are more suitable ; a very
common pair is formed of iron and the alloy const antan.
(For industrial purposes thermo-couples are classified as
" rare-metal " and " base-metal " respectively.)

As already mentioned, the indication of the galvanometer
depends on the difference in temperature between the hot
junction and the cold junction. If ordinary wire leads are
used to connect the thermo-couple to the indicator, the " cold
junction " will be in effect where these leads join the thermo-
couple, and this may be affected by the source of heat. If,
however, the two ends of the thermo-couple are connected
by leads of the same material as the materials of the thermo-
couple itself, the cold junction is transferred to the galvano-
meter, which is some distance away and not likely to be
affected by the source.1 This is usually done in the case of the
iron-constantan couples, but in the case of the rare-metal
couple it would be very expensive to have long leads of
platinum and platinum-rhodium. The difficulty can be got
over by using " Peake's compensated leads," which are wires
of inexpensive alloys chosen to give the same electromotive
force as the platinum — platinum -rhodium. The compensa-
tion cannot be made quite exact for all temperatures.

Thermo-electric pyrometers are on the whole less accurate
than resistance thermometers, but are more convenient to
use and are more adapted to industrial purposes. They can
be used for rather higher temperatures than platinum resis-
tance thermometers and can also be used for low temperatures.

Radiation Pyrometers. — In many cases it is not desirable
or even possible to have an instrument in actual contact with
the hot body ; and for temperatures over 1,500° C. none of
the above pyrometers is suitable. In such cases radiation

1 Another, and more accurate, method, is to have the cold junction
in a Dewar vacuum vessel which keeps its temperature remarkably
constant.

INDUSTRIAL PYROMETRY

£47

pyrometers are used. These instruments make use of the
laws of radiation mentioned on page 241. Most of those used
in practice belong to one or other of two types.

In the one type the intensity of the total radiation is meas-
ured, and the calibration is based on Stefan's Law. In the
other the intensity of the radiation of some particular wave
length is measured or rather compared with that from a
standard source ; as this comparison is carried out by optical
methods it cannot be used for temperatures below about
700° C., i.e. " dull red heat," whereas the total-radiation type

(a) (b)

FIG. 108.

can be used for temperatures as low as 500° C. There is no
upper limit to the temperature which radiation instruments
are capable of measuring.

The best-known instrument of the first type is the Fery
radiation pyrometer (Fig. 108). It is made up of a short
tube open at one end directed towards the hot object. At
a point on the axis of this tube is a small iron-constantan
thermo-couple, the two wires of which are connected to
terminals on the outside through two metal strips A A. The
radiation is focussed on to the thermo-couple by a concave
mirror C, which can be moved by a pinion P. In order to
see whether the focussing is correct an eyepiece E is provided,

248 HEAT

and one of the strips A carries a short tube B in which are
two semicircular plane mirrors inclined at a small angle to
one another and with an opening at the axis of the tube
through which the radiation passes. If the radiation is not
correctly focussed, the general appearance on looking through
the eyepiece is as in Fig. 108 (a) ; the pinion is turned until
the appearance is as in Fig. 108 (6). The electromotive
force generated in the thermo-couple is indicated by a gal-
vanometer connected to the terminals. The tube B and its
mirrors screen off some of the reflected radiation from the
thermo-couple ; the rays which do reach it form a cone of
which the angle is determined by the arrangement of B and
is not altered when the mirror is moved in focussing.

The indications of the instrument are independent of its
distance from the source of heat if the latter is not too great.
This follows from the small area of the thermo-couple exposed
to radiation.

If the instrument is near the hot body the concave mirror
forms a large image of it, and only a small part of the image
covers the thermo-couple ; if the instrument is further away
the image is smaller and consequently a larger part of it is
on the thermo-couple. The effect of distance is thus auto-
matically compensated until the stage is reached that the
image is so small that the whole of it is on the thermo-couple ;
beyond this distance the indications of the galvanometer
would fall off.1

In the case of instruments of the second type, the radiation
of some particular colour is matched against light of the
same colour from a standardized electric lamp. These instru-
ments are usually classified under the heading of " optical
pyrometers." Several very good optical pyrometers are
obtainable, of which we may take the " disappearing filament "
type as an example.

1 Another instrument of this type, the Foster-fixed focus pyrometer,
has a diaphragm near the open end of the tube, and the mirror is arranged
so that the diaphragm and the thermo-couple are at conjugate foci,
In this case no focussing is needed.

INDUSTRIAL PYROMETRY 249

The hob object is viewed through a species of telescope,
the eyepiece of which is fitted with a piece of glass which
transmits only light of one colour, usually red. Inside the
telescope is a small electric lamp ; the position of which is
adjusted so that the filament is in the same plane as the
image of the hot body formed by the object glass. Thus,
on looking through the eyepiece the image of the hot body
appears superposed on the filament. The current through
the lamp is varied until the filament just disappears ; this
means that as far as the light of the one colour is concerned,
the brightness of the filament and image are the same.

Thus, the temperature of the filament corresponds to that
represented by the image, and the current indicator of the
lamp is calibrated to show directly the temperature of the
hot body. As in the case of the other kind of radiation
instrument, the readings are independent of the distance of
the hot body within wide limits.

The Meaning of the Temperature indicated by Radia-
tion Pyrometers. — It was explained on page 241 that the
various radiation formulae, such as Stefan's, Planck's, etc ,
apply only to the case of full radiation or the radiation from
a perfect black body. In the case of bodies which are not
black no very satisfactory quantitative laws are obtainable ;
it is obvious also that if they were we should have a differed
set of values for each surface. Radiation pyrometers are
therefore calibrated on the basis of full radiation and they
give what is known as the " black body temperature " ol
an object, i.e. the temperature of a perfectly black body
which would give the same value of radiation as the actual
body. Fortunately, however, in many of the cases for which
radiation pyrometers are used the conditions approximate
fairly closely to those of a black body. For example, the
case of a furnace where the radiation is observed through a
small opening in the door ; an ingot of metal actually in a
furnace (where it is required to heat the metal to a particular
temperature before withdrawing it for the next process), or
a muffle at a uniform temperature all give practically ful]

250 HEAT

radiation. Again, metals which when heated become tar-
nished with a black oxide approximate to a black body.

On the other hand, if the pyrometer is sighted on a body
not in an enclosure and having a highly reflecting surface,
the " black body " temperature recorded will be considerably
lower than the actual temperature. All that the pyrometer
tells us then is that the temperature of the body is certainly
not lower than a particular value.

It may be mentioned that the radiation from a body with
a reflecting surface is in general less deficient in the regions
ot short wave length than in those of the long ; consequently,
the temperature given by an optical pyrometer is likely to
be nearer the truth than that given by a total-radiation
pyrometer.1

Temperature of the Sun. — By making use of the radia-
tion laws it is possible to obtain an estimate of the temperature

FIG. 109.

of the sun. The greater number of determinations are based
on total radiation. First of all the amount of energy radiated
in unit time on a given area of the earth's surface is deter-
mined. The apparatus used for this purpose is known as a
pyroheliometer. In its earliest form (that due to Pouillet)
this is simply a thin copper calorimeter, blackened on the
surface facing the sun and silvered on the remaining surfaces.
The calorimeter A (Fig. 109) was mounted on an axle B,

1 Standard books on temperature measurement are Ezer Griffith's
Methods of Measuring Temperature (Griffin, 1918) and the somewhat
older The Measurement of High Temperatures, Burgess and Le Chate-
lier. A very complete account (also by Griffiths) of pyrometers is

INDUSTRIAL PYROMETRY 251

on the other end of which was a disc C. By moving the axle
so that the shadow of A just covered the disc, it could be
arranged that the sun's radiation fell perpendicularly on the
blackened face of A. The rate of rise in temperature of the
contents of the calorimeter was noted and also the rate of
cooling when the apparatus was screened from the sunshine.
Knowing the thermal capacity of the calorimeter and con-
tents and allowing for the cooling, the radiation energy falling
per minute on the end of A was calculated. This is not a
very satisfactory method, and a much better form was used
by Abbot and Fowle in 1909 (Smithsonian Reports or Preston's
Theory of Heat). Allowance has then to be made for absorp-
tion in the atmosphere. This is difficult, as the absorption
is different for different wave lengths and is also very much
dependent on the amount of water vapour in the atmosphere.
It may be estimated by comparing the radiation received
when the sun is nearly overhead and when it is near the horizon
so that the radiation traverses a greater thickness of the
atmosphere ; also by comparing the effects at stations of
different elevation.1

Many determinations have been made in this and other
ways, and the values generally regarded as trustworthy range
from 2 to 2-5 calories per minute on each square cm. of the
earth's surface after allowing for absorption in the atmosphere.
We will assume a mean value 2-3 calories per minute and
obtain a value for the sun's temperature. Taking the mean

given in the Dictionary of Applied Physics, Vol. I. (Macmillan). Fury's
original instrument is described in Comptes Rendus, Vol. 134, p. 977
(1902). Preston's Theory of Heat has a good account of the subject.
Much information is to be got from makers' lists, especially those of
the Cambridge and Paul Instrument Company. The lists of Fournier
et Cie of Paris give vapour-pressure instruments, and Negretti and
Zambra the compensated steel- bulb mercury thermometer. Griffin's
catalogue has a good deal about resistance thermometers and thermo-
couples. Reference may also be made to a paper by R. S. Whipple on
" Modern Methods of Measuring Temperature," Proceedings of the
Institution of Mechanical Engineers, July, 1913 ; and to the ever-useful
table of Physical Constants of Kaye and Laby.

1Langley, Researches on Solar Heat, 1884. Phil Mag.t 1883.
Langley and Abbot, Smithsonian Reports, 1903 and later.

252 HEAT

distance of the earth from the sun as 93,000,000 miles and
assuming that the sun is radiating equally in all directions,
this means that each sq. cm. on a sphere of radius 93,000,000
miles is receiving energy at the rate of 2-3 calories per minute.
The energy radiated from the surface of the sun per minute
must be equal to the energy falling per minute on the surface
of any sphere which surrounds it. Thus the energy radiated
from each square cm. of the sun's surface must be —
2-3 X area of surface of a sphere of 93,000,000 miles radius

Area of the sun's surface
calories per minute.

Taking the sun's radius as 430,000 miles, this is —

2-3 X 47i x (93,000,000)2_ 2-3 X (9-3)2 X 1014 calories per

4ji X (430,000) 2 (4-3) 2 X 1010 minute

_ 2-3 X (9-3)2 X 1014 calories per

(4-3)2 X 1011 x 6 second.

Now by Stefan's Law the energy radiated by a black body
is—

Aa04 or 06* per unit area.

a has been determined by experiments of the kind performed
by Lummer and Pringeheim (page 240). The recent deter-
mination by Millikan (1917) gives a as 5-7 X 10"5 ergs per
second. Taking 1 calory as 4-2 x 107 ergs, we have —
2-3 X (9-3)2 X 4-2 x 1021 = 04 where 6 is the black-body
(4-3) 2 X 6 X 5-7 X 106 temperature of the sun.

0 = 6,003° A.

Thus the black-body temperature of the sun is 6,000° A. or
5,730° C.

It should be noticed that this is the black-body temperature ;
the actual mean temperature would probably be somewhat
higher. Most of the recent determinations of the radiation
energy received at the earth's surface are nearer to 2 calories
per minute.

Other estimates of the sun's temperature are based on the
distribution of energy in the spectrum.

The general conclusion to be drawn from the various
experiments is that the black-body temperature of the sue

INDUSTRIAL PYROMETRY 253

is somewhere between 5,500° and 6,000° Absolute, probably
nearer the latter.

Calibration of Pyrometers — Meaning of a Scale of
Temperature. — It is important to realize what is meant by
a scale of temperature and to what extent a temperature
measurement is reliable ; it is particularly important in the
case of high temperatures. When we speak of a temperature
of 0° C. or of 100° C. the meaning is perfectly clear ; we mean
the temperature at which pure ice melts or the temperature
at which the vapour pressure of pure water is one standard
atmosphere.

But the statement that the temperature of a substance is
50° C. or 150° C. has no such definite meaning..

In fixing our scale of temperature we take some particular
substance (e.g. mercury in a glass container), find out how
much some arbitrary property, such as its volume, changes
when the temperature is raised from 0° C. to 100° C. and divide
that change into 100 equal parts ; a rise in temperature of
1° is then defined as such a change of temperature as will
bring about a change equal to one of these parts. 50° C.
then means such a temperature that when the particular
substance is heated from the melting-point of ice to that
temperature, the expansion is half what it would be if it
were heated from the melting-point of ice to the boiling-point
of water under standard pressure.

It does not follow that the scale of temperature defined in
terms of one substance will agree with a scale defined in terms
of another (except, of course, at 0° C. and 100° C.) ; and in
fact they do differ. In the case of a number of suitable
substances such as mercury,1 alcohol, or a gas, the scales of
temperature do not differ among themselves by very much
between 0° and 100° C., but at high temperatures (alcohol is
not available) the difference may be a matter of several degrees.
In the case of the not easily liquefiable gases such as hydrogen

1 It is clear that a substance such as water, which can have the same
volume at two different temperatures, is unsuitable for a thermometrio
substance

254 HEAT

or nitrogen, especially if the temperature be defined in terms
of the change in pressure at constant volume, the agreement
between the scales is found to be very close ; 1 these gases
are therefore very suitable for thermometric substances.
They have the further advantages that they can be used
over a very wide range of temperature and the actual change
in pressure per degree is relatively large.

In practice, therefore, the hydrogen scale or the nitrogen
scale are taken as the basis of temperature measurement ;
other instruments, such as mercury thermometers, thermo-
couples, or platinum resistance thermometers, are standardized
by comparison with a constant volume hydrogen thermometer
or a constant volume nitrogen thermometer. From a theo-
retical standpoint the hydrogen thermometer is the better ;
but at high temperatures trouble is experienced owing to
diffusion of hydrogen through the walls of the bulb. For an
accurate standard the bulb should be of metal and is usually
of platinum or platinum iridium ; at temperatures above
400° C. hydrogen diffuses through this, so that for high
temperatures a -nitrogen thermometer is used. For very
low temperatures and for temperatures up to 400° C. the
constant volume hydrogen thermometer is the basis of tem-
perature measurements ; above this temperature the constant
volume nitrogen thermometer is the basis, and accurate
standardizations have been carried out with it for tempera-
tures up to 1,600° C.2

When temperatures are measured with various instru-
ments they can all be expressed in terms of the hydrogen
scale or the nitrogen scale if they are not above 1,600° C. ;
such temperatures have a definite meaning apart from the
particular instrument used to measure them. But when we
come to measure temperatures such as 2,000° C. with some

1 Observe that this is not taking the ft coefficient as the same for
each gas, but finding the actual pressure at 0° C. and 100° C. in any
given case and taking 1° as the rise in temperature required to produce
,-£?> of the above change in pressure.

* Cp. Swann, Carnegie Institute Publications.

INDUSTRIAL PYROMETRY 255

form of radiation pyrometer, the result means that we have
determined the behaviour of the pyrometer at temperatures
below 1,600°C. and assume the law to hold for higher tem-
peratures. Such temperatures cannot therefore have the
same definite meaning as the others ; the measurements of
the same temperature by different types of pyrometers may
differ by a matter of a few degrees.1

Finally, it may be pointed out that there is a theoretical
scale of temperature — the absolute thermodynamic scale
(Chapter XXI) — which does not depend on the properties of
any substances. Equations have been reduced, based on the
porous plug effect, which enable relations to be expressed
between temperatures on either hydrogen or nitrogen scale
and temperatures on this thermodynamic scale.

1 Much information on this subject is given in Ezer Griffith's book,
which has already been mentioned. References to original papers will
be found in it. See also Dictionary of Applied Physics.

PART II

CHAPTER XVII
THE PROPERTIES OF GASES

In Chapter VI it was stated that Boyle's Law was not strictly
true for any gas, especially at high pressures ; it was further
pointed out that some gases obeyed the law much more nearly
than others. A rather more detailed study of this departure from
the Law will throw a good deal of light on the
properties of gases.

The deviation is very marked in the case of
the gas sulphur dioxide ; so much so that it
may readily be illustrated by a simple experi-
ment. A and B (Fig. 110) represent two tubes,
closed at the top by taps, and joined at their
lower ends where they are connected by a long
piece of rubber tubing with a mercury reser-
voir C which can be raised 'or lowered. B is
filled with sulphur dioxide and A with air ; the
quantity of air is adjusted so that when the
level of the mercury in B and C is the same, the
mercury in A comes to the same height. Thus
at the start the air and sulphur dioxide are at
the same pressure. The reservoir C is then
gradually raised so that the gases are com-
pressed. So long as the pressure is not much
above the atmosphere the mercury in A and B
rises about the same amount, but as C is raised further the
mercury in B rises above that in A, showing that the sulphur
dioxide is more compressible than the air. The difference be-
comes more marked until a stage is reached when the pressure
is something over 2 atmospheres (if the room is fairly cold — at

257 8

FIG. 110.

258 HEAT

0° C., a pressure of just over 1£ atmospheres would suffice), at
which there is a quick rise of the mercury in B and the sulphur
dioxide condenses to a liquid.

Other gases, such as chlorine or carbon dioxide, behave in a
similar way, but in their case a much greater pressure is required
to produce liquefaction ; an apparatus of the type shown in
Fig. 110 would not be very suitable for showing this effect (apart
from the fact that chlorine has a chemical action on mercury) as
the reservoir C would have to be raised to inconvenient heights
to obtain sufficient pressure to bring about liquefaction. Never-
theless the general effect is the same, i.e. as the pressure is raised
the gas becomes more compressible than it would be if it obeyed
Boyle's Law (so that the product pv instead of remaining con-
stant becomes smaller), and when a certain pressure is reached,
depending on the nature of the gas and its temperature, the gas
begins to liquefy, and if the pressure is maintained at this value
the whole of the gas turns to liquid.1

On the other hand no such effect is found with gases like air,
nitrogen or oxygen. Careful experiments show that they do
deviate slightly from Boyle's Law ; Regnault showed that at high
pressures they do become slightly more compressible than they
would be according to Boyle's Law, i.e. pv diminishes.

The process, however, does not continue indefinitely ; a series
of experiments by Amagat * showed that as the pressure increases
the product pv at first diminishes, but that at still higher pres-
sures pv begins to increase again and reaches a value greater than
it was at the beginning. In the case of hydrogen the experi-
ments of Regnault and the later experiments of Amagat showed
that the product pv increased from the start as the pressure
increased ; there was no initial stage when the gas was
more compressible than Boyle's Law would indicate.3

1 Accurate experiments show that the pressure has to be raised
slightly above that at which liquefaction commences in order to com-
plete the process. This slight increase has an important bearing on
the theory of the subject.

2 Amagat, Ann. de Chimie et de Physique, 1881.

3 Casual inspection of the reproductions of Amagat's diagrams
sometimes produces rather exaggerated ideas of the magnitude of these
effects. As a rough indication the following figures may be quoted.
At 15° C. or 16° C. increasing the pressure from 1 to 25 atmos-
pheres would produce a decrease in the value of pv of well under 1 per
cent, for nitrogen, about 1 per cent, for air, something like 25 per

PROPERTIES OF GASES

259

Thus at first sight there appears to be a fundamental difference
between a gas like carbon dioxide and a gas like nitrogen. But
the meaning of this difference becomes clearer when the effect of
temperature is considered. It is found that the pressure neces-
sary to liquefy any particular gas increases as the temperature is
raised, and that when the temperature exceeds a certain value,
known as the critical
temperature of the gas,
it is impossible to bring
about liquefaction how-
ever great a pressure is
applied. This point may
be well illustrated by
considering the classical
experiments performed
by Andrews x with carbon
dioxide. These consisted
in observing the changes
in volume of the gas as
the pressure is increased,
and repeating the opera-
tion with the gas main- n \] Waier^ |— ^ U*f-Hfcfcr
tained at temperatures
which were successively
increased.

The gas was contained
in a glass tube of the
form shown in Fig. Ill

(1). The part AB was of capillary bore, BC was about 2£ nun.
bore and CD half this. The bore of AB was calibrated by
introducing a thread of mercury and measuring its length at
different positions and weighing the mercury so introduced. At
the beginning both ends of the tube were open, and carbon

-Air

cent, for CO2 and an increase of about 1 per cent, for hydrogen. At
60 atmospheres pv for nitrogen would have passed its minimum value
(1 per cent, decrease) and would be increasing, for air it would have
decreased about 2 per cent, and CO2 would have liquefied. At 300
atmospheres the value of pv for nitrogen would be about 20 per cent,
above its original value, but the density of the gas would be compar-
able with that of a liquid, being something like -3 gm. per 0.0.
*Phil Trans., 1869.

260 HEAT

dioxide was drawn through for a considerable time to get rid of
residual air,1 the end A was sealed off and D closed ; D was then
placed under mercury and opened. By gently warming the tube
some of the gas escaped, and on cooling mercury entered the tube.
The final quantity of gas was adjusted by placing the tube (with
its end still under mercury) under the receiver of an air pump and
reducing the pressure until such an amount of gas escaped that
on restoring the pressure the mercury entered to the desired
height. The glass tube was then enclosed in a stout copper tube,
Fig. Ill (2); this communicated with a similar copper tube
containing a glass vessel similar to ABCD (1) but containing air.
The copper tubes were completely filled with water and were fitted
with screw plungers ; by screwing in either or both of these
plungers pressure was transmitted to the two glass tubes. The
object of the air tube was to measure the pressure, which was
calculated from the compression of the air. At that time Amagat
had not made his experiments, and so the pressures quoted by
Andrews are based on the assumption that air obeys Boyle's
Law ; he had no reliable data for any correction. The gases
could be kept at any desired temperature by means of a jacket.

A series of experiments were made at different temperatures
and from the results curves were plotted. The general form of
these curves is shown in Fig. 112, which is not drawn to scale.
Each curve represents the behaviour of the gas at one particular
temperature, and is spoken of as an isothermal curve.

Referring to the curve at 13-1° C. represented by ABCD.
From A to B the volume of the gas diminishes as the pressure
increases, and the general form of the curve is not dissimilar to
the corresponding curve for air at 13'1° C., except that the value
of pv is less than for air. When the pressure reaches the value
represented by B (about 49 atmospheres) liquefaction begins and
continues without increase of pressure until all the gas is lique-
fied,* a state represented by C. The liquid being much less com-
pressible than gas, the curve rises steeply to D, a large increase of
pressure producing small changes of volume. Nevertheless the

1 In spite of these precautions traces of air remained which had some
effect on the results.

2 Strictly speaking, it was found that BC was not parallel to the
volume axis, an extra pressure of about J atmosphere being required
to complete the liquefaction. This effect was largely due to residual
air already referred to.

PROPERTIES OF GASES

261

liquid in the neighbourhood of C is much more compressible than
ordinary liquids. In the next curve, corresponding to 21-5°, a
much higher pressure is reached before liquefaction begins, and
the horizontal portion is shorter than CB. The gas being com-
pressed to a smaller volume before liquefying,1 there is less change
in volume in passing to the liquid ; moreover the liquid at
21-5° C. occupies a perceptibly larger volume than the liquid at

-700

-50

13-1°

Volume.

FIG. 112.

13-1° C. ; the coefficient of expansion of liquid CO2 at pressure
just enough to keep it liquid is very large.

As the temperature is raised the change in volume in passing
from gas to liquid becomes smaller, the horizontal portion of the
curve becoming shorter. At a sufficiently high temperature the
horizontal portion vanishes altogether ; this is represented by
the curve for 31-10. There is no stage at which decrease in

1 It must be remembered that the curves refer to the same mass of
CO, in every case.

262 HEAT

volume occurs while the pressure remains stationary ; all that
happens is that there is a change in the steepness of the curve, but
it does not become parallel to the volume axis. It was found that
there was no formation of liquid ; no separation was noticed, and
the appearance of the gas did not change. There is a rapid
change in density at this point but no separation. Repeated
experiments showed that the highest temperature at which it was
possible to get the CO2 visibly to form liquid was 30-92° C. This
temperature is therefore the critical temperature for carbon
dioxide. At 32-5° C. there is still an inflection of the curve, but
it becomes less marked, and at 48*1° C. the curve shows no abrupt
change in volume at any pressure, but is similar in appearance to
the corresponding curve for air.

Thus when the temperature is well above the critical temper-
ature the behaviour of carbon dioxide is very similar to that of
air or nitrogen. This at once suggests that the properties of air
or nitrogen are due to the fact that at ordinary temperatures they
are much above their critical temperatures, and this is found to
be the case.

The critical temperature of nitrogen is —146° C., and so it is
not surprising that it was found difficult to liquefy it. Still more
difficult is it to liquefy hydrogen, of which the critical temperature
is — 234-5° C. The gas helium long resisted all attempts to
liquefy it, and it was not until 1908 that Kamerlingh Onnes
succeeded in doing so. As the critical temperature of helium is
somewhere about —268° C., or 5° above absolute zero, it is not
surprising that difficulty was experienced. A reference to the
methods available for liquefying gases will be found later (pages
265 and 289).

Gases and Vapours. It has already been mentioned that
sulphur dioxide at a temperature of 0° C. will liquefy under a
pressure of about 1-5 atmospheres. If however the temperature
be lowered to —10-1° C., the gas will liquefy under ordinary
atmospheric pressure.

Now if we have some water vapour enclosed in a tube and kept
at atmospheric pressure it will remain vapour as long as the
temperature is above 100° C., but if the temperature falls below
100° C. the vapour will condense, since the vapour pressure of
water below 100° C. is less than 760 mm. of mercury. The two
cases are precisely similar ; below — 10-1° C. the vapour pressure

PROPERTIES OF GASES 263

of sulphur dioxide is less than 760 mm. It is usual to talk of
sulphur dioxide as a gas and water vapour as a vapour, but the
distinction is very arbitrary and arises merely because we hap pen
to live in a climate where the temperature is above the boiling
point of sulphur dioxide and below that of water ; it would be
more logical to regard sulphur dioxide as an unsaturated vapour
rather far removed from its saturation pressure. If we wish to
make a strict distinction between a gas and a vapour the critical
temperature will serve as a criterion ; if the substance is below its
critical temperature it can be liquefied, and if it is above it cannot.
Thus hydrogen above —234-5° C. may strictly be called a gas ;
below that temperature it would be an unsaturated vapour at
atmospheric pressure until the temperature fell below — 252'7° C.
the boiling point of hydrogen.

Water above 100° C. and under atmospheric pressure is an
unsaturated vapour, but if the temperature is above 365° C. it
passes its critical point and becomes a gas which cannot be
liquefied.

Further Note on the Critical Temperature. — Turning again to
the diagram (Fig. 112) of Andrew's curves for carbon dioxide,
there is one point in the interpretation of them which has not yet
been considered. In the lowest isothermal, the volume occupied
by the substance at B when it is all gas and is just about to liquefy
is very much greater than at C, where it has all just become liquid.
Thus the density of the gas is much less than that of the liquid
On the next isothermal the corresponding volume change is less •
the difference between the density of gas and liquid is less. This
difference would diminish for each succeeding isothermal, and just
below the critical temperature the difference would be almost
nothing, the density of gas and liquid being almost the same.
This general effect, which is not confined to carbon dioxide but
is characteristic of all substances at this stage, might be weD
illustrated by an experiment of the following type.

Let Fig. 113 represent a thick-walled glass tube containing
some liquid.1 Let the space above be free from air and contain
nothing but the vapour of the liquid (this could be arranged by
starting with the tube open at the top, boiling the liquid for some
time to drive out air, and then sealing off). The pressure of the

1 It is desirable not to use water since it attacks glass at high tem-
peratures.

264

HEAT

FIG. 113.

vapour will correspond to the saturation pressure at the particular
temperature. Now let the temperature be raised. The satu-
ration pressure being greater, some evaporation will take place and
the density of the vapour will increase. At the same time the
liquid will expand owing to the increase in tempera-
ture, and so will become less dense. Thus as the tem-
perature rises the density of the vapour increases, and
that of the liquid decreases. This will go on until,
when the temperature is just below the critical tem-
perature, the density of the vapour will be very nearly
equal to that of the liquid. The boundary between
the two will become less distinct,1 and finally as the tem-
perature passes the critical point the densities become
equal, the boundary disappears and the tube is full
of one homogeneous substance ; the distinction be-
tween liquid and vapour has vanished. Thus the
critical temperature may be regarded as the limiting
temperature when owing to the density of the liquid and its
saturated vapour becoming the same the distinction between
them vanishes.

This way of arriving at the idea of the critical temperature way
in fact the historical way in which the concep-
tion came. The classical experiments were per-
formed by Cagniard de la Tour.1 His method
was similar to that of the preceding paragraph
except that he was able to measure the pressure
at the same time. The liquid and vapour were
contained in the end A (Fig. 114) of the bent
tube which contained mercury ; the space B con-
tained air, which by its compression indicated
the pressure. These experiments were performed
many years before those of Andrews ; shortly
afterwards Faraday succeeded in liquefying a
number of gases and considerably extended the
knowledge of the effect of pressure and low tem-
perature on a gas.

The idea that the distinction between gas and liquid may

1 The surface tension of a liquid diminishes as the temperature rises,
so that the meniscus will become flatter ; at the critical temperature
the surface tension becomes zero.

2 Annales de Chimie et de Physique, 1822-23.

FIG. 114.

PROPERTIES OF GASES

265

Air under
-pressure

Liquid

disappear is further strengthened when we consider that it is
possible to pass from an undoubted liquid to an undoubted gas or
vice versa without it being possible to say that the change has
occurred at any particular stage. For suppose we have a vessel
as in Fig. 115, with the space A filled with
a liquid, while B contains air under very
great pressure. Let the liquid A be heated
until its temperature is above the critical
point ; if the pressure is great enough, no
change takes place except a slight expan-
sion. Now let the pressure in B be re-
leased. The substance in A will simply
expand like any other gas ; there will be
no change from liquid to vapour. Thus
the substance started as a liquid and
finished as a highly compressed gas and at
no time was there any abrupt change.

Liquefaction of Gases. — It is clear that
if a gas is at a temperature lower than its
critical temperature it can be liquefied by ex-
posing it to sufficient pressure. Gases like sulphur dioxide, chlorine,
ammonia, and carbon dioxide, if cooled by means of freezing
mixtures, can be liquefied without requiring extravagantly large
pressures. (Sulphur dioxide liquefies at atmospheric pressure
when cooled below — 10° C.) Such gases when liquefied can be
used to obtain much lower temperatures by allowing the liquid so
formed to evaporate rapidly under reduced pressures and produce
cooling by evaporation. This can then be used as the starting-
point in the liquefaction of other gases, and was in fact so used in
the early experiments on the liquefaction of oxygen. But other
methods are necessary in the case of hydrogen, and even for
oxygen are far more effective. We shall consider these modern
methods in Chapter XIX.

FIG. 115.

CHAPTER XVIII
THE KINETIC THEORY OF GASES

Having considered a few of the outstanding properties of gases,
we must now turn to the attempts which have been made to
frame a theory of the matter. The object of such a theory is to
help us to form a mental picture of what is going on so as to co-
ordinate the various observed effects and to anticipate what is
likely to happen in any given conditions. From very early times
the phenomena of compressibility of matter in all its states,
solution of substances in liquids, diffusibility of fluids and the
expansibility of gases have led to the idea of matter being con-
stituted of particles with more or less space between them.
Various chemical phenomena led to the idea of these particles as
molecules each made up of a system of one or more atoms of the
same or different natures according to whether the substance is
an element or compound, and recent physical work has shown
that the atoms themselves are complex systems and atoms of
different elements have certain constituents in common. For our
present purpose we shall not need to consider anything beyond
the molecule, although in one point (see page 305) we shall have
occasion to refer to the number of atoms in the molecule.

The characteristic property of a solid is its ability to retain its
shape apparently for an indefinite period. This suggests that the
molecules are not able to move about from one part of the sub-
stance to another. Moreover solids possess cohesion — a great
deal of work is necessary to tear one portion of a solid away from
another. This suggests that the molecules exert considerable
force upon one another. The phenomena of expansion on heating,
thermal conductivity, and some connected with radiation,
suggest some movement of the molecules, and we may regard each
of them as capable of vibrating about some mean position but not
of permanent displacement. In the case of liquids, such pheno-
mena as diffusion suggest chat the molecules can move about

266

KINETIC THEORY OF GASES

267

freely from one part to another ; on the other hand, a liquid has
a definite volume at any particular temperature and may also
(when free from dissolved gases) show very considerable cohesion.
We must therefore regard the molecules as still under the influence
of each other's action

A gas has the property of indefinite expansion and hardly
appreciable cohesion (but see page 285 on the porous plug effect).
We may therefore regard the molecules as moving about freely
and for the most part not under the influence of each other's
action. The enormous extent to which the volume can be
reduced by pressure suggests that the molecules only occupy a
very small fraction of the whole volume of the gas. This is the
starting-point of the kinetic theory.

On this theory an ideal gas is regarded as made up of a very
large number of particles (molecules) of negligible size compared
to the volume of the gas, moving about in all directions with
considerable velocity and constantly bombarding the walls of the
enclosure and colliding with one another. At these collisions no
loss of energy occurs, a molecule rebounding from the walls with
a velocity equal to that with which it encountered them. The
pressure exerted by the gas is re-
garded as due to this bombardment.
Any mutual attractions between
the particles are also supposed to
be negligible.

Consider a cubical enclosure of
side e (Fig. 116) containing a gas.
Let there be N molecules each of
mass m.

Suppose that any particular mol-
ecule has a velocity u in any
direction. We can resolve this velocity into three components
x, y, z, parallel to three adjacent edges of the cube. Since these
directions are mutually at right angles, we have by the ordinary
resolution of vectors,

»* + y* +z* =t*» . . . (1)

We can deal with each of these components separately. Con-
sidering the pair of faces of the cube perpendicular to the a; axis,
a molecule starting at one face will travel to the other, bound
back and arrive back at the face from which it started in the

FIG. 116.

268 HEAT

rt A

time it takes to travel 2s with velocity x, i.e. in — seconds. It

•27

will therefore collide with this face -5— times per second.1

At each collision its momentum is changed from mx in one
direction to mx in the opposite, i.e. the change of momentum

£M

is 2mx. Since this occurs -x— times per second, the rate of

change of momentum is „- x 2mx = . But the rate of

change of momentum is equal to the force exerted on the face.
Thus on each pair of faces perpendicular to the x axis there is a

mxz

force exerted due to this molecule of .

s

In the same way on each of the faces perpendicular to the y

my*

axis there is a force , and on each of the other pair the

s

mz2
force is .

S

The total force exerted on all the faces is thus
2mx2 2my2 2mz2 2m(x2 + y* -f 2*) 2mwa

-7- + ~^~ + — " ~T~ ~ = ~r bv W-

The force will not, of course, be the same on each face (unless
it so happened that x = y = z).

If another molecule had a velocity wlf the total force on all

2mult
faces exerted by it would in the same way be - — . Thus

s

the total force exerted by the N molecules would be
- (uz + u^ + w2» + . .)

8

where u, ulf u2, etc., are the respective velocities of the N mole-
cules.

Let U2 be the mean value of u2 + u^ -f- .

1 The argument is not affected by the fact that it may collide with
another molecule on the way. Suppose the second molecule happened
to be at rest when the collision occurred. It would move off with
velocity x and the first would come to rest. Thus a molecule would
still go on with velocity x even though it were not the same one. The
resultant effect is the same whatever the velocities provided no energy
is lost in collision.

KINETIC THEORY OF GASES 269

N.2mU»
Then the total force is - - .

o

Now although the individual molecules are moving with
different velocities and different directions, when we are dealing
with the statistical result of a very large number, there is no
reason for the force in one direction to be greater than that in
another, and we may regard the force as evenly distributed over
the six faces.

We thus have a force of - - distributed over an area of

6s2. The force per unit area (i.e. the pressure) is therefore
2NwUa 1 NrnU2

^r--6*' = 3^--

But s* = v the volume of the cubical enclosure. -

or pv = y NmU» ..... ..... . (2)

We will now show how the various gas laws follow from this
theory. In the first place, it will be evident that U depends on
the temperature. For the kinetic energy of a molecule of the
gas is £mu2, and the total kinetic energy is £NmU*. If the
molecules are of negligible size and have no attractions on each
other they cannot have energy of rotation or energy of position
relative to each other.

If we supply energy to the gas and raise its temperature, the
increase in energy must be represented by increased kinetic
energy, and this means that U must increase. On this theory,
then, increasing the temperature means increasing U ; further-
more if U = 0, i.e. the molecules just stay in a heap, the tem-
perature must have got to the lowest possible limit, i.e. Absolute
Zero.

Boyle's Law — In equation (2) we have

pv =-

9

Now Nra is the total mass of the gas. Thus pv = £MUf -^o
M is the total mass. But if the temperature remains constant

l~\3 is known as the root mean square velocity.

270 HEAT

U remains constant. Thus for any given mass of gas at constant
temperature

pv = constant.

Charles's Law. — If we consider that the addition of equal
amounts of energy produces equal rise of temperature (i.e. the
Thermal Capacity of the gas remains constant at all tempera-
tures), then the internal energy of a given mass of gas will be
proportional to its absolute temperature. Thus |NraU8 « T,

but pv = gNwU2.

pv *> T or pv = RT where R is a constant.
Graham's Law of Diffusion. — Since

pv =-i-NmUs

— --MU 2 where M is the total mass of gas,

But — is the mass per unit volume or the density.
If p is the density we have therefore

P—gpU'.

This means that at any particular pressure the value of pU1
is the same for all gases, or that

U2 is proportional to — .

Graham found that the rates of diffusion of gases were inversely
proportional to the square roots of the densities.

Avogadro's Hypothesis. — Suppose we have two gases, each
at the same pressure p, and let n^ be the number of molecules

per unit volume,
fTij be the mass of each molecule,
H! be the root mean square of the

velocity
for the one gas, and n2, ra2, U2 the values for the other gas.

Then p — -TT n^m^J^.
But p = -7j-n2m2U2*.

KINETIC THEORY OF GASES 271

Thus r^mjlV = n^n^z* ......... (3)

Now if the two gases be allowed to mix, and have no chemical
action on each other, there will be no evolution or absorption of
heat. This means there is on the whole no communication of
energy from one gas to the other ; each set of molecules remains
as before.

But it is a proposition in statistical mechanics, known as
Maxwell's Law of equipartition of energy, that when a system in
equilibrium is made up of a number of components, each com-
ponent being able to possess one kind of energy, the total energy
of the system will on the average be equally divided amongst the
components. In this case the components are the nx molecules
of one kind and the n2 molecules of the other, each of which can
possess kinetic energy of translation. Thus on the average a
molecule of one kind will have the same energy as that of another,
thus^mjUV = iWU* ........... (4)

It follows from (3) and (4) that
n, = n2.1

Influence of the Action of one Molecule on another. — The
kinetic theory which has just been outlined deals with the case
of an ideal gas in which the influence of one molecule on another
is negligible. The phenomenon of cohesion (to say nothing of
any others) indicates that when the molecules are fairly closely
packed they do have a decided action on each other, and so it
is not surprising that actual gases show some deviation from
Boyle's Law and other simple relations, especially when they are
highly compressed.

We will now consider how the simple kinetic theory can be
modified so as to take account of this effect and be brought more
into accordance with observed results. The first really satis-
factory modification is due to Van der Waals in 1879.

There are two aspects of this effect to be considered. First

1 It should be noticed that this demonstration of Avogadro's Hypo-
thesis is based on a piece of statistical mechanics (Maxwell's Law) which
involves conceptions far beyond the scope of this book. It is a funda-
mental proposition of the classical Newtonian mechanics, but experience
is necessary to decide when the conditions are such as to make it safe
to apply it.

272 HEAT

the actual size of the molecules diminishes the space available for
movement, and collisions with the bounding walls are therefore
made more frequent. Thus if we imagine a molecule A (Fig. 117)
travelling from one wall of the vessel to the opposite, it is clear
that the distance it has to go is not the width of the vessel, but
that width minus the diameter of the
• * . molecule. In the same way if the

P~y £N molecule B collides with a molecule C

which is at rest, C moves off with a
I corresponding velocity and hits the op-
FIG. 117. posite wall, so that the effect is the

same as if B had gone on ; only C

starts off not from the point where B steps but one diameter to
the right of it. This general effect may be summed up by saying
that the volume available for movement is not the observed
volume v of the gas but v— b where b is a space proportional to
the volume occupied by the molecules themselves.1

Secondly, the force exerted by one molecule on another. Co-
hesion suggests that there is an attraction between them which
is considerable when the molecules are close together ; but the
behaviour of gases suggests that the force becomes small when
the distance is large. In the present state of our knowledge we
cannot say very much about the nature of this force, but for the
moment we may consider that when two molecules are within a
certain distance of each other there is a force of attraction be-
tween them, but beyond this distance the force is negligible. We
may express this by saying that each molecule has a sphere of
influence of a certain radius ; any molecule which comes within
this sphere will suffer attraction.

The general effect of this is that in addition to the external
pressure tending to prevent the molecules passing out in all
directions there is a mutual attraction between the molecules.

At any given moment this attraction will only be operating on
any molecules which happen to be within the sphere of influence
of another ; the total effect will therefore be proportional to the

1 It might be thought at first sight, that 6 is equal to the volume of
the molecules ; detailed consideration shows that this is not so. Van
der Waals concluded that b was four times this volume, and Jeans has
arrived at the same conclusion. An elementary treatment (which,
however, requires following with care) is given in Poynting and Thom-
son's Heat, page 153.

KINETIC THEORY OF GASES 273

frequency with which such close approaches happen. Now on
the ordinary laws of probability the chance of one particular
molecule coming within a certain distance of some other molecule
is proportional to the number of such molecules per unit volume ;
the chance of any molecule coming within this distance of any
other is therefore proportional to the square of this number.

The frequency of such approaches will therefore be propor-
tional to -j, and so to get the total restraining influence we must

1 »
add to the pressure a term proportional to -^ •

Instead of putting the simple relation

pv = RT
Van der Waals puts

\p -\ — 2J (v —b) = RT where a is a constant.

It should be noticed that as far as deviations from Boyle's Law
are concerned the two effects operate in contrary directions.
For if, instead of pv = constant
we put p(v — b)= constant,
we have pv = pb + constant,

in other words, pv increases with pressure.

On the other hand, if instead of pv = constant

we put (p -f- ~~z)v = constant,

a

we have pv = constant — -

and therefore pv diminishes as v diminishes or p increases. What
actually happens will depend on which factor is of more import-
ance.

In the case of hydrogen where pv increases with pressure, it
appears that the attraction between the molecules is cf small effect

1 This was not the way in which Van der Waals arrived at the term
?. in his expression, but it appears to the writer to have some advantages,

and in particular to make it clearer why the same term — is not to be

expected to express the relation when the compression is so great that
the molecules are practically always within each other's sphere of
influence.

274

HEAT

compared with the other factor. (Cp. page 289.) In gases like air
and nitrogen, however, the effect of the attraction is at first pre-
dominant and pv diminishes ; when the compression becomes
sufficiently great the size of the molecules as compared to the
total volume becomes the predominant factor.

Van der Waals' equation is a much closer representation of the
observed facts in connection with gases than the simple equation
pv = RT. In the case of some substances (halogen derivatives
of benzene) it is obeyed fairly closely both in the gaseous and
liquid states. In the case of other substances, however, it does

not very well represent
the facts for both states.
This is not very surpris-
ing, since in the liquid
state the molecules are
probably within the
sphere of one another's
influence, and in that
case, as pointed out in
the footnote, page 273,
one would not expect
a simple constant a to
represent the facts.
Until we know more
about the nature of
molecular forces it is
not likely that a satisfactory formula can be found. In the
meantime Van der Waals' equation represents a very valuable
help in visualizing what is happening and is undoubtedly on
the right lines.

' There is one aspect of Van der Waals' equation which is of
considerable importance.
If we take the equation

Volume
FIG. 118.

- 6) = RT

it will be seen that it is a cubic equation for v. For it may be
written

(pv2 + a) (v - b) = RTva

or pv 3 — (bp + RT)?;2 + av — ab = 0.

av ab
+ P ~ P

0.

KINETIC THEORY OF GASES

275

Now a cubic equation has three roots, of which either one or
three are real.

This means that for any given value of p and T there are either
one or three possible values of v.

Taking the case of an isothermal for a substance below its
critical point (Fig. 118).

For any value of p above the point A there is only one possible
value of v, and similarly for a point below A. But for the value
A the volume can be vx (all liquid) or v2 (all vapour), or we
could have a mixture of the two at some intermediate value.

Now from quite differ-
ent considerations, based
partly on the fact that it
is possible to cool a vapour
some way below its con-
densing point (in the ab-
sence of dust or any suit-
able condensation nuclei)
and also to heat a liquid
above its boiling point,
James Thomson suggested
that ideal form of the
isothermal might be a con-
tinuous curve of the form
shown (Fig. 119), where the

dotted part represented an unstable state. In this case the third
root of our equation would be represented by vs. This is purely
hypothesis. What is of real meaning is that at the critical tem-
perature, the points vlt v2, v3 coincide. But in this case the value
of p and T must be such that the equation is of the form of a
perfect cube. If pe and T« are the values of the critical pressure
and temperature, then

a ab

/, c/3 «/£

FIG. 119.

must satisfy the condition

(v-ve)* = 0.
Equating coefficients we have

RTe

276

HEAT

p-

ab

PC

Thus b

RT 8a

RTc = 276

Thus there is a definite relation between the critical data of any
substance and the value of the constants a and 6.

Now a and 6 can be deduced from observations of the com-
pressibility of a gas such as those of Amagat. Using these values
the critical temperature can be calculated. 1 The agreement with
the observed value is fairly good in some cases, but in others it
is not so good.

Note on Evaporation — It is interesting to consider the behaviour
of a liquid from the point of view of the kinetic theory. Let A
(Fig. 120) represent a molecule well within the liquid. It is acted

A.

©

B

FIG. 120.

upon only by those molecules within its sphere of influence, so
that if it is in the position A it will suffer attractions in all direc-
tions. These on the average will balance and so there will be no
resultant force tending to prevent it moving. But if it gets to
the surface as at B the available attractions will be downwards,
and to get out from the surface it will have to overcome this
attraction. Thus the molecules in the surface layer will be under
the constraint of forces which tend to keep them in the liquid.
This constraint manifests itself in the phenomenon known as
surface tension.

Now the molecules in the liquid are moving about, and if one

1 Cp. the work of Kammerlingh Onnes in calculating the conditions
necessary to effect the liquefaction of helium, p. 292.

KINETIC THEORY OF GASES 277

comes to the surface it may be going fast enough to get out ; if
it is not it will be retained. The escape of the molecules repre-
sents evaporation, and it will be obvious that the higher the
temperature of the liquid the greater will be the number of fast
moving molecules and therefore the greater the rate of evapor-
ation.1

If the space above the liquid is enclosed the molecules forming
the vapour will be moving about in this space, and some of them
may strike the surface of the liquid and so pass into it again.
The number of those which do so will be greater the greater the
amount of vapour in the space. Finally a state of equilibrium
will be reached when the number of molecules escaping from the
liquid per second is equal to the number which re-enter it. This
is the condition when the space is said to be saturated.

It is clear that since the rate of evaporation increases with
temperature the pressure of the vapour in the space will haye to
be greater at high temperatures in order that the number of
molecules entering the liquid may balance those which get out.
Hence the saturation pressure of a liquid will increase with
temperature.

1 There is a definite connection between the latent heat of evaporation
of a liquid and its surface tension. Cp. Hammick, Phil. Mag., August,
1919, " Latent Heat and Surface Energy."

CHAPTER XIX

PROPERTIES OF A GAS— ENERGY CONSIDERA-
TIONS—LIQUEFACTION OF GASES

Work done by a Gas Expanding. — If we have a quantity of
gas under a given pressure and the gas is made to expand, e.g.
by heating it, work is done against the external pressure. To
illustrate this point let us suppose that we have some gas in a

cylinder fitted with a piston
r* fl£~~*"| which works without friction

(Fig. 121), and is exposed to
the external pressure p. Let
the area of the piston be A ;
then the external force acting

~ ", on it is pA. Now let the gas
expand and push the piston
out through a distance d ; the work done against the external
force is pA. x d or pAd. But Ad is the increase in volume of the
gas and so the work done is p(v2 — Vj), where v2 is the final
volume and vt the initial volume. We have imagined the pre-
sence of a piston in order to fix our mind on the enclosed gas
and see what it is doing, but it is clear that the gas has to do
this work against the external pressure whether the piston is
there or not, since it has to make the extra room v2 — v^ for
itself against the external pressure.1

In many practical cases the pressure doee not remain constant

1 It should be noticed that the shape of the enclosure occupied by
the gas does not affect the work done ; for if the volume has reached
the value vz in any way no work is done in altering the shape at constant
volume since any force, however small, will alter the shape of a fluid.
We can see this is so by imagining a number of side tubes fitted with
pistons. If we push one in the others must move out in such a way
that the increase in volume due to the pistons moving out is equal to
the decrease due to those moving in, and the work done in the two
cases is p x v and — pv respectively.

278

PROPERTIES OF A GAS

279

as the gas expands. If, however, we know how the pressure varies
we can still find the work done. Let us suppose that the pressure
and volume of a gas are represented by the curve AB, Fig. 122 ;
for example, the gas may be kept at constant temperature, in
which case AB is the ordinary isothermal curve. Consider the
work done as the volume increases by a series of small stages.
Let the first stage be from v^ to X represented by the points A
and C on the curve. If the pressure at A is pt and the pressure
at C is pg, we shall only make a small error if we take the work

done as (X — vt) x average pressure or

Pi ~f~ P*

— v,). This

is represented on the diagram as the area of the rectangle HEXt^

Similarly the work

done from X to Y

will be represented

by the area LMYX.

The work done from

vl to v2 will there-

fore be the sum of

all the rectangles

shown by the dotted

lines ; that is to

soy, it is the area

of the figure

y z

Volume
FIG. 122.

If the different stages
are taken much
smaller, the zigzag

line HKLM . . . . U will become nearer to the curve AB, and
if the steps are made indefinitely small the zigzag line will, in the
imit, coincide with AB. Thus the work done is actually repre-
sented by the area ABv2vlt that is to say by the area bounded
by AB, the two ordinates at vv and v2 and the axis of volumes.
This process is only a graphical representation of the process of
integration, so that in the notation of the calculus we can
say

Work done against external pressure = I pdv.

280 HEAT

Thus so long as we know the form of the curve representing the
relation of pressure and volume we can calculate the external
work done.

When the gas expands, the energy to do this external work has
to be supplied from somewhere ; if it is not supplied, and some
compressed gas is allowed to expand against external pressure,
the energy to do the external work is taken from the gas itself,
which becomes cooler. For example, the thermal capacity of a
given mass of gas will be greater if we keep the pressure constant
and let the gas expand as its temperature rises than it will be if
we keep the volume constant ; in the one case external work is
done and in the other it is not. In speaking of the specific heat
of a gas it is therefore necessary to state the conditions ; we speak
of the specific heat at constant pressure and the specific heat at
constant volume respectively ; the ratio of these two specific
heats is usually denoted by 7. The value of y depends on the
constitution of the gas and plays an important part in problems
connected with the energy ; we shall see, in the succeeding
chapter, that it can be determined without finding the values
of the specific heats separately. It is, however, possible to
find either specific heat, and the methods available are as
follows.

Determination of the Specific Heat at Constant Pressure. —

The determination of the specific heat at constant pressure of a
number of gases was successfully carried out many years before
the difficulties of the constant volume determination were
overcome.

Regnault l carried out a number of experiments, and his
method may be indicated as follows : —

The general idea was to pass the gas through a heating coil,
whereby its temperature was raised to some particular value, and
then to let it flow through a thin metal vessel contained in a
calorimeter and so measure the heat given up. The pure dry gas
was contained in a large reservoir A (Fig. 123), which was pro-
vided with an open manometer B. A was kept at a constant
temperature by a water bath, and the amount of gas which had
passed out was determined from the reading of the manometer
before and after the experiment. The volume of the reservoir

1 Mimoirea d' Academic de* Sciences, vol. 26, 1862.

PROPERTIES OF A GAS

281

being known and the temperature constant, the quantity of gas
was calculated from the fall in pressure.1

The gas passed through a regulating valve C ; beyond this
valve was a water manometer G, and C was gradually opened as
the experiment proceeded, so that the pressure indicated by G
remains constant. Thus although the pressure in the reservoir
was falling the rate of flow of gas remained steady, and the
pressure at which it entered the calorimeter was constant.

The gas passed through a spiral tube of thin copper, immersed
in a vessel of oil D, heated by a spirit lamp. This tube was so
long (about 10 metres of tubing) that by the time the gas reached
its lower end it had taken up the temperature of the oil bath.1

Fio. 123.

The gas then entered the calorimeter E ; here it passed through
a vessel made of thin brass shown diagrammatically in the
sketch ; actually it consisted of a number of chambers with par-
titions, so that the gas flowed against a large surface of the walls

1 To make sure of the accuracy of this method, a subsidiary experi-
ment was performed in which the escaping gas was collected and weighed
and the weight compared with that calculated from the manometer
readings.

2 Subsidiary experiments were made with a thermometer in the tube
to make sure that the gas did take up this temperature.

282 HEAT

of the vessel. Finally it escaped to the atmosphere at F. A
thermometer at F showed that the temperature of the escaping
gas was the same as that of the calorimeter. If T is the tem-
perature of the gas entering the calorimeter and ^ and t% the
initial and final temperatures of the calorimeter and contents,

the average fall in temperature of the gas is T — 1 2 2, and hence

the heat given up by the gas is ms\T — 2).

In order to prevent, as much as possible, transference of heat
from the heater to the calorimeter by conduction along the tube
the connecting part was made of non-conducting material. The
heater and lamp were surrounded by an outer vessel to diminish
radiation to the calorimeter, and the latter was enclosed in a
wooden box, one side of which is represented at D (Fig. 123). In
making the experiment, observations were first made for an
interval of ten minutes, without having any gas flowing, to
determine the rate at which the calorimeter received heat by
conduction and radiation ; the gas was then turned on and the
rise in temperature of the calorimeter and its contents noticed ;
finally the gas was stopped and the rate of change of temperature
of the calorimeter noted ; in this case we have the cooling effect of
the calorimeter to the surroundings superposed on the heating
effect due to conduction and radiation from the heater.

The difficulty of this experiment lies in the fact that the gas
must pass through fairly slowly in order to be sure that it finally
attains the temperature of the calorimeter ; but this means that
the passage of gas must go on for some time if sufficient mass is
to pass through to give a reasonable rise in temperature. The
various corrections therefore represent an important fraction of
the whole effect, and it is in the evaluation of these corrections
that the trouble comes in. It is a tribute to the skill of Regnault
that his results agree as well as they do with those obtained by
later and improved methods.1

Specific Heat at Constant Volume. — In this case the gas must
be enclosed in some vessel, and the mass of this vessel will be
large compared to that of the gas. The relatively large thermal

1 A fairly full account of later work, especially that of Lussana. is
given in Preston's Theory of Heat, 3rd edition. Mention should also be
made of the work of Swan (Proc. Roy. Soc., Dec., 1908) arid others.

PROPERTIES OF A GAS

283

capacity of the container presented great difficulties, and it was
not until the introduction of the steam calorimeter by Joly that
satisfactory direct determinations were made.1 For this purpose
Joly * used a modified form of his apparatus which he called a
differential steam calorimeter.

Two equal copper spheres (Fig. 124), 6-7 cm. in diameter,
hung from opposite arms of the balance in the same steam
chamber. Catchwaters A A were fitted to catch any of the con-
densed steam which might drop off the spheres. The two spheres
with their catchwaters were made of equal thermal capacity. One
sphere was empty and the
other contained the gas under
experiment at a considerable
pressure — in some cases 20
atmospheres or more. The
excess of condensation on the
one side over that on the other
gave the condensation due to
the contained gas ; the effect
of the thermal capacity of the
containers and their buoyancy FIG. 124.

effect (see page 169) were thus

eliminated. The great merit of this arrangement is that the
quantity of heat absorbed by the gas as its temperature is raised
is directly determined ; it does not appear as the difference
between two large quantities separately determined.

SPECIFIC HEATS OP GASES BY DIRECT DETERMINATION.

Constant pressure.

Constant volume.

Hydrogen ....
Nitrogen
Air . .

3-40
•23
•24

2-40
•17
•17

Carbon dioxide .
Argon

•20
•12

•16

•075

1 Up to this time the value of the specific heat at constant volume
was calculated from that at constant pressure by the methods indicated
on page 298.

* Phil. Trans., 1891.

284 HEAT

Mayer's Calculation of the Mechanical Equivalent of Heat
from the Values of the Specific Heats of a Gas. — If the

difference between the specific heats of a gas is solely due to
the fact that when the gas expands it does work against the
external pressure, we can obtain a value for the mechanical
equivalent of heat since the coefficient of expansion of the gas is
known. This was suggested by Mayer x in 1842 ; he took the
case of air and worked out a value for J. At that time, however,
the available data in connection with specific heats of gases were
very unreliable, and the value he obtained is of little interest.
We may illustrate the method by taking modern results ; it will
be convenient to take the case of hydrogen.

If we take the weight of 1 litre of hydrogen at 0° C. and
760 mm. of mercury as 0-0899 gms. the volume of 1 gm. of

hydrogen at this temperature and pressure is TTogq litres or

11,130 c.c.
Taking the value for the coefficient of expansion of hydrogen

as 27"o (CP- PaSe 75), the increase in volume of 11,130 c.c. when

heated from 0° C. to 1° C. is ~^^~ c.c. The pressure is 76

cm. of mercury or 76 X 13-6 X 981 dynes per sq. cm.
The work done is therefore p x (v2~'yi)
76 x 13-6 x 981 x 11,130

273

This work accounts for the difference between the thermal
capacity of 1 gm. of hydrogen at constant pressure and constant
volume respectively. This difference is 3-40—2-40 = 1 calorie.
Thus 1 calorie is equivalent to
76 x 13-6 x 981 x 11,130

Internal Work when a Gas Expands. — The method of calcu
lating J which has just been given depends on the assumption
that when a gas expands the only work it has to do is in pushing
back the external pressure. If, however, there is a mutual attrac-
tion between the molecules work will also be done in pulling
them further apart ; this may be spoken of as the internal work.
Mayer assumed that no internal work was needed ; Joule, who

1 Liebig'a Annalen. 1 842.

PROPERTIES OF A GAS

285

began his work on the mechanical equivalent of heat in 1840,
made a number of experiments in connection with this point.
Two copper reservoirs A and B (Fig. 125) were connected by a
pipe fitted with a tap. A contained dry air at a pressure of 22
atmospheres, and B was exhausted. The two reservoirs were
placed in a calorimeter. On opening the tap air rushed from A
to B until the pressure was the same in each ; no external work
was done since the gas does not have to push back the atmo-
spheric pressure. On the other hand, if there is a mutual attrac-
tion between the molecules work will have to be done in increasing
the distance between them, and the gas will tend to absorb heat
from its surroundings to supply
the necessary energy. Joule was
not able to detect any fall in tem-
perature of the water in the calori-
meter although his thermometers
read to 55° F., and the calorimeter
was shaped so that the quantity
of water required to cover the
reservoirs was as small as possible.
The experiment was tried with A
<md B in separate calorimeters,
in which case there was a distinct
cooling effect in A's calorimeter
and a nearly equal heating effect
in B's. These experiments indi-
cated that the internal work was inappreciable and appeared
to justify Mayer's assumption. But the thermal capacity of
the reservoirs and the calorimeter were so large in comparison
with that of the gas that unless the cooling effect was consider-
able it was likely to escape detection. All that the experiments
showed was that there was no large effect.

A more sensitive method was proposed by Sir W. Thomson
(Lord Kelvin), and he and Joule carried oub a series of experi-
ments in 1852 which gave decisive evidence of internal work.
These experiments are of fundamental importance (see pages
319 and 290) and must now be considered.

The Joule -Thomson Porous Plug Experiments.1 — The general

1 Thomson, Math, and Phy. Papers, vol. 1. Joule, Scientific Papers
vol. 2. Phil. Trans., 1853 and 1854, 1862.

FIG. 125.

"I'll'l

286 HEAT

idea in these experiments is to cause the gas under pressure to
flow along a tube in which there is a very constricted opening at
one point which results in a considerable drop in pressure ; such
an arrangement would be referred to by engineers as throttling.
Let Fig. 126 represent a pump driving gas along the tube
in which there is a narrow constriction. Eddies are formed by
the gas rushing through the constriction, but these subside a
little beyond the opening as their energy is frittered away. Let

us suppose that the
tube is of large dia-
meter so that the
velocity of the gas is
very small except near
the opening and its
j2 W., kinetic energy negli-

* — ^ - gible, and let a steady

FIG. 126. state have been

reached. Let pt be

the pressure in the tube on one side of the opening and pQ that
on the other ; if the tube is open to the atmosphere (as it
was in the actual experiments) pQ will be sensibly equal to the
atmospheric pressure (since the velocity of the gas is small).

Suppose that in a certain time a mass of gas m has passed
through the opening. Let v^ be the volume occupied by this gas
under a pressure plt and VQ its volume under pressure p0. The
passage of this gas into the low pressure side of the tube means
an increase in volume v0 on this side and involves pushing back the
atmospheric pressure; the work done is therefore p0vQ (page 2 78).
But the volume of gas in the high pressure side has diminished
and the pump has therefore done an amount of work on the gas
aqual to Piv^. If the gas obeys Boyle's Law, pjVt = p0v0.

The amount of work done on the gas is therefore equal to that
done by it, and the gas does not have to draw on its own energy
to do the work of expansion.

Thus as far as external work goes there would be no reason for
any change in temperature of the gas.

But if work is necessary to separate the molecules against their
mutual attraction some energy must be supplied to do the neces-
sary work in expanding from vl to v0. This additional energy
would have to be supplied by the gas itself, and so the tempera-

PROPERTIES OF A GAS

287

ture would fall. It was this effect that Joule and Thomson
looked for.

One modification of the arrangement of Fig. 126 was necessary.
The gas issuing from the hole has a large velocity and therefore
kinetic energy. If the total energy of the gas is not altered it
must have less energy of other kinds and its temperature would
therefore be lower.1

Further on where the eddies have subsided this kinetic energy
has been reconverted to heat and the temperature would be
restored. It would not do, therefore, to seek for a cooling effect
by taking the tempera-
ture close to the orifice,
while further on the
gas has had a chance of
receiving heat from the
walls of the tube and
other sources, and a
small effect would be
missed. For this reason
Joule and Thomson re-
placed the simple aper-
ture with a porous plug
of cotton wool (in some
cases silk fibres) con-
fined by two perforated E-E\E
brass plates (Fig. 127).
In this way the gas
which issued possessed
hardly any kinetic energy. The gas was passed at a slow
steady rate through a long spiral copper tube immersed in a
water bath at constant temperature. The upper end of the
tube terminated in a short boxwood tube containing the plug. A
thermometer was suspended immediately above the plug and a
portion of the tube was made of glass to enable the thermometer
to be read. Surrounding the boxwood tube was a metal jacket
filled with cotton wool, to protect the tube from the influence of
external heat, in contact with the boxwood tube. The gas was
allowed to pass for a considerable time (about an hour) before

Boxwooti
Tube

Copper
Tube

FIG. 127.

1 A large effect of this kind was in fact found by Joule and Thomson
in a subsidiary experiment with a simple orifice.

288 HEAT

taking any readings in order to allow initial fluctuations of tem-
perature to subside and make sure that a steady state had been
reached.

They found that in the case of all gases except hydrogen there
was a cooling effect, which was much more marked for carbon
dioxide than for air, oxygen or nitrogen. Hydrogen, however,
gave a slight but distinct heating effect. In all cases the thermal
effect was proportional to the difference in pressure on the two
sides of the plug.

At first sight the behaviour of hydrogen appears to be very
extraordinary, but in the interpretation of these results there is
one point to be remembered. In the simple treatment of page
286, it was pointed out that if the gas obeys Boyle's Law no
change in the energy of the gas is brought about by the external
work done ; the work done p^v^ on the gas (the energy supplied
by the pump) being equal to the work p0vQ done by the gas.

But no gas obeys the Law absolutely, and if p^v^ is not equal
to pQvQ there is a balance of external work to be considered. For
pressures such as were used in this experiment the value of pv for
all gases except hydrogen diminishes with pressure, i.e. the gases
are more compressible than Boyle's Law indicates. Thus ptvt
would be less than PQVO, and so the work done on the gas would
be less than the external work done by it ; the energy necessary
for the balance of work done by the gas would be drawn from the
gas itself, and there would be a cooling effect on this account.
This cooling effect can be calculated for any gas by using Amagat's
values for the variation of pv. When this is done it is found that
it is very much less than that observed by Joule and Thomson, so
that there still remains a decided cooling effect due to internal
work. l

In the case of hydrogen pv increases with pressure, and thus
piPi is greater than p0v0, so that the external work done on the
gas is greater than that done by it. There is thus a heating effect
due to this cause. When the value is worked out from Amagat's
value for the deviation from Boyle's Law it is found to be about
the same magnitude as that observed by Joule and Thomson.
It follows therefore that in the case of hydrogen at moderate

1 For example, in the case of air, the cooling due to this deviation
from Boyle's Law only amounts to a little over T'n of the total effect
observed.

LIQUEFACTION OF GASES 289

pressures the mutual action of the molecules upon one another
is very small indeed.

Temperature Inversion of the Cooling Effect. — Joule and
Thomson found that the cooling effect diminished as the initial
temperature of the gas was increased. Later work has shown
that for every gas there is some temperature at which the cooling
effect becomes zero, and that above this temperature a heating
effect is observed. Hydrogen, as we have seen, gives a heating
effect, but at sufficiently low temperatures it is found to give a
cooling effect.

Olszewski found that the temperature of inversion for hydro-
gen was —80-5° C. ; below this temperature there is a cooling
effect and above it a heating. This point is of great importance
in the liquefaction of hydrogen.

Liquefaction of Gases.

The work done by an expanding gas is the basis of the successful
methods of liquefying the more permanent gases. In the early
work the gas was compressed to a very high pressure in a vessel
cooled by suitable refrigerators such as liquid carbon dioxide, and
the pressure then suddenly released, e.g. by opening a tap. The
gas rushed out, doing work against the external pressure and
thereby became cooled ; if the initial pressure were high enough
some of the gas liquefied. In this way oxygen was first liquefied
in 1877 by Cailletet and Pictet, working independently.

The method was developed, and a better refrigerating substance
tor the initial cooling was rapidly evaporating liquid ethylene.
Oxygen, nitrogen and air were thus liquefied, and about 1883
Wroblewski succeeded in liquefying hydrogen, and later Olszewski
obtained an appreciable quantity.

Modern Methods. — The method described above is not suit-
able for the production of considerable quantities ; some form of
continuous process is needed. The introduction of the regener-
ative method of cooling marked the beginning of a new period in
the history of the subject. The method may be indicated by the
following outline of a process suitable for a gas such as air.

The air is compressed to a very high pressure by a pump A
(Fig. 128) and passes through a coil of copper piping B in a water
tank whereby the heat generated in compression is absorbed and
the gas passes on, at atmospheric temperature through a long

U

290

HEAT

copper spiral C. At the lower end is a nozzle D, the opening of
which is controlled by a valve.

There it expands down to atmospheric pressure and cools. The
cooled gas sweeps up past the coil and thereby cools the next
portion of gas coming down inside the coil. Successive portions
of the oncoming gas are cooled further and further until at last,
if the initial pressure is sufficient,1 some of the gas liquefies and is

FIG. 128. •

collected in a vessel F. The coil is surrounded by a vessel E of
good heat insulating material.

The conditions under which the expansion is done are quite
different from those of the older method of Pictet and others.
The compressed gas is being driven by the pump along the system,
and hence although work is being done at the nozzle in expanding
against atmospheric pressure, the energy to do it is mainly
derived from the gas behind, which in turn is receiving it from
fche pump. The conditions are therefore almost similar to those
of the porous plug experiment, and the cooling effect is chiefly
due to the Joule Thomson effect of internal work. This effect,
which is almost insignificant at moderate pressures, becomes
quite large at high pressures and low temperatures.

1 A pressure of about 200 atmospheres or a little less is a typical
pressure for liquid air apparatus.

LIQUEFACTION OF GASES

291

This point becomes clearer when we consider the arrangement
for gases other than air. It should be noticed that the gas
coming down the spiral C is cooled by the expanded gas passing
up outside it ; the latter is therefore receiving heat, and its tem-
perature rises as it passes up E. There must be a sufficient length
of tubing in C to enable a suitable exchange of heat to take place,
and when the apparatus is well designed and working properly the
rate of flow of gas is such that its temperature, as it issues from
E, is not much below that of the atmosphere. If we are dealing
with gases other than air, which have to be prepared beforehand,
there is no reason why the issuing gas should not be taken from
E back to the compressor (as indicated by the dotted line exten-
sion to E) and used over again, for we are relying on the Joule
Thomson effect and not on the external work for .the coolings.
This is in fact done.

The credit of making use
of the Joule Thomson effect
is generally attributed to
Hampson and Linde, who
first independently ob-
tained liquid air without
the use of refrigerating
agents. Much useful work
was done by Dewar, and
his vacuum vessels (the
origin of thermos flasks)
were a most valuable con-
tribution to the problem
of heat insulation and the
handling of liquefied gases.
Kammerlingh Onnes also
did much work at this
time, and his later work,
including the liquefaction
of helium (page 292), is
classical.

Liquefaction of Hydro-
gen.— Since the Joule
Thomson effect for hy-
drogen above —80-5° C.

FIG. 129.

is positive, i.e. a heating effect, it is

292 HEAT

clear that we cannot begin with hydrogen at atmospheric tem-
perature and expect to liquefy it by this process which depends
on internal work. In order to do so the hydrogen must first be
cooled well below the inversion point by external means. For
this purpose liquid air is used, and the plant shown in Fig. 128 is
modified by the addition of the necessary cooling coils. Fig. 129
shows diagrammatically the arrangement used by Travers. The
cooling is done in stages. The compressed hydrogen (at 200
atmospheres) passed through the coil B, which was immersed in
a mixture of solid CO2 and alcohol. It then passed through a
coil G surrounded by a vessel which was kept filled with liquid
air, and from G to a coil H in a vessel in which liquid air was kept
boiling under reduced pressure. The gas was thus cooled to
about —200° C. It then entered the regenerator coil C and
escaped from the nozzle D. The expanded gas passes up over the
regenerator coil and round the outside of the vessels surrounding
G and H and so back to the supply main and the compressor.
The nozzle opening was varied by a specially constructed valve
worked by a rod passing down the centre of the coils and fitted
•with a milled head K. The liquid hydrogen collected in the
vessel F. Very complete heat insulation was required through-
out.

Improved forms of the apparatus have since been designed in
which the coil B is cooled by the returning hydrogen after it has
reached L.

Liquid hydrogen boils at about —253° C. under atmospheric
pressure. By causing it to boil under reduced pressure Dewar
succeeded in obtaining solid hydrogen. Solid hydrogen melts at
-259° C.

Liquefaction of Helium. — At the temperature of liquid
hydrogen all other substances have become solid with the single
exception of helium. This substance remains a gas even when
compressed and cooled by rapidly boiling liquid hydrogen (i.e.
below — 260° C. ) and then suddenly allowed to expand. For a long
time therefore it defied all attempts to liquefy it. Kammerlingh
Onnes spent many years working at the problem. He first of all
set to work to find out accurately such matters as the critical
temperature and the inversion point of the Joule Thomson effect ;
for helium, like hydrogen, has a positive value for this effect at
ordinary temperatures. He did this by determining a large

LIQUEFACTION OF GASES 293

number of isothermal curves at low temperatures and so deducting
the values of a and b in Van der Waals' equation. By this means
he calculated the critical temperature and also the inversion point,
which can be found from a knowledge of the positions of the
minimum value of pv.

The critical temperature was uncertain, as different isother-
mals did not indicate the same value, but he satisfied himself
that at any rate the inversion point was above the temperature
of rapidly boiling liquid hydrogen, and so concluded that it ought
to be possible to apply the regenerative method successfully.

He designed and made an apparatus on the lines already indi-
cated, using rapidly boiling liquid hydrogen as the refrigerating
agent. At 7.30 p.m. on July 10, 1908, liquid helium was seen
for the first time, and something like 60 c.c. were obtained in this
original preparation. The best commentary on this work is to
give Onnes' later values for the constants of this gas : —

Critical temperature, — 268°C.

Boiling point (760 mm.), -268-8° C.1

1 The standard book on the liquefaction of gases is Travers' Study of
Gases. A brief account of the liquefaction of helium is given in Nature,
August, 1908. Much work has been done in recent years and it is to be
regretted that there is no recent edition of Travers' or some similar book.
Much work on liquefaction was done during the war in the efforts to
obtain helium in quantity from certain natural gases for the purpose
of filling large airships with this light non-inflammable gae.

CHAPTER XX
THE GAS EQUATIONS

The experiments of Joule and Thomson indicate that at ordi-
nary temperatures and pressures the internal work done during
expansion by a gas like hydrogen or nitrogen is very small ;
experiments such as those of Amagat show that in these con-
ditions of temperature and pressure the deviation from Boyle's
Law is also small.

If then we assume that a gas exactly fulfils Boyle's Law and
that the internal work during expansion is zero, the results we
deduce will not be far from the truth when applied to actual
gases, and if we require greater accuracy a correction can be
applied by using the results of the porous plug experiment and
those of Amagat.

A gas which fulfils the two conditions is spoken of as a " perfect
gas."

The general equation for a perfect gas is pv = RT where T is
the absolute temperature and R a constant. This equation is
not the expression of a physical law (other than that of Boyle),
but is really only a definition of temperature on the gas scale.
The physical law, that of Charles, is that temperatures defined in
this way by means of different gases will agree closely with one
another. We can apply the reasoning of page 280, to determine
the meaning of R. Suppose we have a mass of gas at a tempera-
ture T, and it is heated at constant pressure p to a temperature
T + t. If CP be the thermal capacity of the gas at constant
pressure the heat required will be CP x t. During this process the
volume will increase to v + v' where v' is given by the relation
p(v + «') = R(T + t)

But pv = RT

. • . pv' = R£,

pv' represents the external work done and so Hi is equal to the
external work.

294

THE GAS EQUATIONS 295

If the gas were heated at constant volume, no external work
would be done and the heat required is C. X t. Since we
suppose there is no internal work, the whole of the difference
between Cp x I and C. x / is accounted for by the external
work.1

If J is the mechanical equivalent of heat, we have
J(Cp x t -C. x t) = R*
or J(C, - Cr) = R.

Thus R represents the external work done per unit rise in
temperature at constant pressure ; it is equal to J times the
difference between the two thermal capacities if these are
expressed in ordinary heat units.

It is expressed as so many units of work per degree per unit
mass of gas and is thus a constant for each gas, and is spoken of
as the gas constant. For example, since 1 gm. of hydrogen
occupies 11,130 c.c. at 760 mm. and 0° C. R for hydrogen

11,130 x 76 x 13-6 x 981
is — 273 — —ergs per gm. per degree centigrade.

If the mass is in pounds and the pressure in pounds weight per
sq. foot and the volume in cubic feet the value of R would be in
foot-pounds per Ib. per degree centigrade. If, however, we take
the gram molecule of the gas as our unit the value of R comes
out the same for all perfect gases (see page 303).

Isothermal and Adiabatic Changes — The general equation
pv = RT represents the connection between pressure, volume
and temperature of a gas and holds good for all possible variations
in condition which a gas may undergo. There are, however,
certain particular changes where a special relation holds good in
addition to the general equation, and the two most important are
isothermal and adiabatic.

Isothermal Changes. — If the temperature remains constant,
the relation between p and v is given by Boyle's Law, i.e. pv =
constant. This is a particular case of the general equation.
Since work is done by the gas when it expands it follows that
during an isothermal expansion heat must be supplied to the gas
to maintain the temperature at its constant value, and during
isothermal compression heat must be abstracted from the gas,

1 It is assumed that C, is independent of the pressure. In the case
of actual gases Joly found slight variations.

296 HEAT

otherwise the energy supplied to it by the work done during
compression would cause a rise in temperature.

Adiabatic Changes. — If the gas is isolated, and no heat enters
or leaves it during the change the latter is said to be adiabatic.
A moment's consideration will show that the change in pressure
which results from a given change of volume in an adiabatic
transformation will be greater than that which accompanies an
equal change in volume in an isothermal one. For suppose we
have some gas at a pressure p and temperature T l occupying a
volume v. Let it be compressed adiabatically until the volume
is reduced to vf. Work is done on the gas and therefore (since no
heat can leave it) the temperature will be raised. If pf be the
resulting pressure, the product p'v' will be greater than pv since
the gas is now at a higher temperature. Thus p' must be greater
than that which would correspond to the volume v' at the original
temperature. In the same way, if the gas expands the tempera-
ture will fall and the drop in pressure will be greater than that
which corresponds to an isothermal change.

Equation of an Adiabatic for a Perfect Gas. — The relation
between p and v for an adiabatic change is given by the equation
pvy = constant where 7 is the ratio of the two specific heats.
This equation can be deduced from energy considerations in the
following manner : —

If a gas is heated, and at the same time allowed to expand in
any way against external pressure, the amount of heat absorbed
by the gas is equal to C» x rise in temperature + the thermal
equivalent of the external work done. This is a perfectly general
relation whether the pressure remains constant or not. Let us
consider a very small change so that we may write the external
work as p • dv, where dv is the small increase in volume (for a

finite change it would be I pdv).

We can write the general relation in the form

<5H =» C«*5T +-j . pdv where (5H is the heat absorbed

(5T is the rise in temperature
and dv the change in volume.

1 Throughout the remainder of this chapter, whenever temperature
is mentioned it represents absolute temperature unless the contrary is
stated.

THE GAS EQUATIONS 297

For an adiabatic change SB. = 0, and the equation becomes
in this case

0 =O<5T 4-y .pdv ..... (1)

Now the complete relation for a gas is the equation

pv = RT, which covers all cases.

If small changes take place in the variables the relation between
them is got by differentiating the general equation, and we have
for any small change

pdv 4- vdp = R<5T (since R is constant)

pdv + vdp
or dl = - 1^ — -....,. (2)

Substitute this value of <5T in (1) and we have

C 1

0 = j^- (pdv + vdp) + y . pdv . • . . . (3)

But R = J(CV-C.) in all cases (page 295).
Hence (3) can be written

0 -

This form is rather cumbrous ; multiply all through by

J(C, — C.)

- Q- - , which is not zero, and we have

Q _ Q^

0 = pdv 4- vdp -\ -- ^ — - . pdv.

^9
/-I

Now -~- is denoted by y, so that the equation is

0 = pdv + vdp + (y-l)pdv
or 0 = vdp + ypdv.
Thus for a small adiabatic change

ypdv = —vdp. Divide through by pv so as to get a form
which can be intergrated and we have
dv —dp

* *i /r\

<*L_ r &p_

v- -J p

an arbitrary constant

or ylog.v = — log«p 4- a constant
.*. log«(''y) 4- log«p = constant
.*. pvy = constant.

The above equation gives the relation between p and v for an
adiabatic change. The case often arises in which the temperatiire

298 HEAT

variations in an adiabatic change are required. All that is neces-
sary is to make use of the general equation pv = RT, which is
true for all kinds of change. For example, we can substitute for

T?rP

p the value , in which case the equation for an adiabatic

becomes

T>nn

. . v1 = constant,

v

or Tvy~l = some constant (since R is constant).

If the relation for T and p were required, one would merely

T> rr>

substitute for v.1

P

Experimental Determination of y. — The relation for an adia-
batic can be used as a means of determining y ; before Joly's
work the value of Cv was found indirectly by determining Cp and
y. One of the most perfect forms of an adiabatic change is met
with in the propagation of sound waves in a gas. The different
portions of the gas suffer alternate compression and rarefaction,
and if the pitch of the note is fairly high these compressions and
rarefactions take place so rapidly that there is very little chance
for any appreciable transference of heat by conduction.

A knowledge of the velocity of sound gives a relation between
p and v for small adiabatic changes.

It is shown in text-books on sound that the velocity of a com-
pression wave such as constitutes a sound wave is given by

u = A / —
V p

when E is the elasticity of the material for the particular kind of
compression and p is the density.

E is defined as the ratio of the stress to the strain ; in the case
of compression of a gas the stress is the increase in pressure which
we may call dp, and the strain is the fraction of the original value

8v
by which the volume is diminished, or - — . The minus sign

1 In the opinion of the writer it is a mistake for any one to try and
remember a large number of formulae and their variations. Nearly
all the gas problems can be dealt with by means of pvy — constant
(which is the easiest form of the adiabatic equation to remember) and
the general equation pv — RT. It is a poor use for a mind capable
of original thought to load it with memorized formulae.

THE GAS EQUATIONS 299

is used because in the ordinary notation of the calculus dv repre-
sents the small increase in volume, and in our case the volume
diminishes when the pressure increases.

dv dp

Thus E = dp -f- — — or — u. c- which, when the changes are

.dp
vanishingly small becomes — v. -r .

For an adiabatic change we have

pvy = constant.
Differentiating with respect to v we have

or v.

dp _|_ yp = 0.

dv

dp
Thus -vlv = yp.

dp
But E is equal to — V-JT.

Thus for adiabatic compression we have E = yp.
So that as the propagation of sound waves is a case of adiabatic
compressions

To illustrate this method let us take the case of air. Experi-
ments show that for dry air at 0° C.

u = 33,190 cm. per second.
If the temperature is constant p and p vary in the same propor-

p pv
tion, so that u is independent of the pressure (for — = — , where

in is the mass of the gas).

If p = 76 cm. of mercury =76 x 13-6 x 981 dynes per sq.
cm.

1 This result represents a solution due to Laplace of a classical
difficulty in the theory of sound which remained unsolved for many

/ "R1
years. Newton obtained the general result u = \J — for the velocity

and then deduced the value for K by assuming Boyle's Law, which gave
E = p. The velocity of sound thus calculated came out very much
lower than the observed values, and the result was an unsolved mystery
until Laplace pointed out that the compressions were adiahatic.

300 HEAT

1 litre of air weighs 1-293 gms. . ". p = -001293 gms. per c.c.

Thus 33,190 =\/y

x 13-6 x 981

•001293
(33,190)a X -001293
"' y : 76 x 13-6 x 981

= 1-406.

The methods of determining the velocity of sound in air are
described in elementary text-books of sound ; in the case of other
gases it is usual to obtain the ratio of the velocity of sound in the
gas to that in air by the Kundt's tube method.

Example of the Use of the Adiabatic Relation. — A type of the
problem which sometimes arises in practical work (and perhaps
even more frequently in examination questions) is as follows : —
A quantity of air, for which 7 is about 1-41, is suddenly com-
pressed to one-third of its original volume. If the initial tem-
perature of the air is 15° C., find its temperature after compression.
Treating the compression as adiabatic, we can make use of the
relation Tv?""1 = constant.

If v is the original volume of the air, we have the initial tem-
perature as 288° A.

y-J

<Vir;

v=T = 288 . v^~l

288 X v*— l = constant =Tl x (— ,

3

or Tx = 288 X 3y~]
.'. log T! = log 288 + -41 log 3

= 2-6550
T, = 451-9° A.
= 178-9° C.
Thus the temperature after compression is about 179° C.

In another type of problem, the pressures might be given.
Suppose we have some air at 15° C. and atmospheric pressure,
and it is compressed adiabatically until the pressure is 2 atmo-
spheres.

Taking the equation just given Tvy—1 and substituting from

T> <y

the general equation - - for v,

We have T. ^ — J = constant or T7^1""7 = constant.

.'. 288Y

THE GAS EQUATIONS

301

\288y

Or288

= 2'

- ; log 2

logTj - log 288
.'. logTi-2-4594 = -3010--2133 '
logT1 = 2-5471

Ta = 352-5° A. = 79-5° C. »

Some other Methods of Determining y. — We must now refer
briefly to three methods of determining y, due respectively to
Clemont and Desormes, Lum-
mer and Pringsheim, and
Jamin and Richard.

The method of Clemont and
Desormes is of interest histori-
cally as one of the earliest
attempts to measure y. A
modified form of the experi-
ment is as follows : —

A large glass globe A (Fig.
130) is fitted with a wide stop-
cock B and a narrow mano-
meter tube C, containing a
liquid of very low vapour pres-
sure and preferably of low den-
sity ; Fleuss pump oil satisfies
these conditions. A contains
the dry gas at a pressure
lightly above that of the external atmosphere. The gas
having taken up the temperature of its surroundings, the reading
of the manometer is taken. B is then opened so that some gas
escapes from A and the pressure falls to atmospheric.1 B is then

1 It is instructive to compare this heating effect of adiabatic com-
pression with the Joule Thomson effect. With air at this temperature
the cooling effect for a drop from 2 atmospheres to 1 atmosphere
was about a quarter of a degree. Thus although at high pressures and
low temperatures the .Joule Thomson effect is sufficiently important to
be the basis of gas liquefaction, at ordinary temperatures and pressures
it is relatively very small and the error introduced by neglecting it is
not serious.

2 In Clemont ana Desormes' method for air the initial pressure was
below atmospheric so that air came in when B was opened. This is

FIG. 130.

302 HEAT

immediately closed. The expansion having been rapid, is prac-
tically adiabatic, and the gas in A cools slightly. On being left
for a time it receives heat from its surroundings, and the tem-
perature returns to the original value ; the pressure therefore
rises slightly and the final reading of C is noted.

Consider the history of that portion of the gas which does not
escape.

Initially it occupies some volume v rather less than that of the
apparatus ; it expands adiabatically to V (that of the apparatus),
and finally it receives heat and returns to its original temperature
but continues to occupy the volume V (we neglect the small
change in volume due to motion of the liquid in the manometer).

Let pl be the initial pressure, p2 the final pressure, and P that
of the atmosphere.

In the adiabatic expansion we have

pX-PV* (1)

In the beginning, the gas occupies a volume v at pressure p:, and
at the conclusion, when it has returned to its original temperature
it occupies a volume V at a pressure p2.

Hence p:v = p2V (2)

p /V\y
Thus from (1) we have -p = ^—

p. V
and from (2) £ =-

Pi

p

log p,-log P = y(log PJ- log p2)-
log PI— log P

7 "log Pi -log P2

The method is open to various objections. Unless the stop-
cock is very wide the outrush of gas sets up oscillations and the
final state depends on the stage the oscillations have reached
when the stopcock is closed. The expansion is probably not
truly adiabatic, as the gas is in contact with a large surface of
glass ; it is noteworthy that even in the improved methods the
value of y comes out low in the case of hydrogen, which is a much
better conductor than air.

open to serious objections (see Poynting and Thomson's Heat). The
method of starting with air at high pressure was first used by Gay
Lussac and Welter.

THE GAS EQUATIONS 303

i

C16raont and Desormes' results were low.

The method has been improved by Maneuvrier * and by
Rontgen,1 who obtained a good value in the case of air.*

Lummer and Pringsheim.* — This is a modification of the
above, in which the fall in temperature of the gas was measured
when it expanded adiabatically from pl to P. The calculation
for y then follows as on page 300, from the equation
TYp1~Y= constant.

Jumin and Richard ' adopted the method of supplying by
electrical means the same quantity of heat to two equal masses
of gas ; one was kept at constant pressure and the expansion
noted, the other was kept at constant volume and the increase
of pressure noted.

They obtained the values 1-41 for air and for hydrogen.

On the whole it may be said that the most reliable methods of
obtaining '7 are: —

(1) Direct determination of CP and C9 separately.

(2) Velocity of sound.

Relation of y to the Composition of a Gas. — We can obtain
some insight into this problem by considering the theoretical
value of y on the Kinetic Theory.

First, apart from any question of the kinetic theory, we have
the relation of p. 295, that

J(C, - C.) - R.

Let us take as our unit of mass the gram molecule of the gas.
To get the value of R in this case we have : —

Molecular weight in grams of any gas at 0° C. and 760 nun.
occupies 22-4 litres.

Hence from the relation pv = RT we have: —

76 x 13-6 x 981 x 22,400 = R x 273

R = 9-318 X 107 ergs per degree.

Now J = 4-18 X 107 ergs per calorie.
8-318 X 107

~

4-18 x 107
= 2 calories (approx.).

1 Comptes Rendus, 1895. * Pogg. Ann., 1873.

* This sentence shows the view taken by the writer as to the general
reliability of the method : it is assumed that the result obtained by
other methods is the test of this experiment.

4 Ann. der Physik, 1898. 6 Comptf* Rendus, 1870.

304 HEAT

If we regard the gas from the kinetic theory and assume that
the molecules are simply infinitesimally small particles, the only
energy they can have is kinetic energy of translation. Thus if
we heat the gas at constant volume, i.e. so that no external work
is done, the heat absorbed (C, x rise in temperature) all goes
in increasing the kinetic energy of translation. The kinetic
energy of each molecule is £mu2, so that of the gas is £Nmwa
(page 269).

On the kinetic theory

pv = g Nrau1.

Therefore kinetic energy of the gas is equal to

3

I*"

Let us take the molecular weight in gms. of a gas at 0° and
760 mm. and raise its temperature from 0° C. to 1° C. at constant
volume.

The gain in kinetic energy will be
3
n (change in pv)

3 1
= -^- x -g—r- X (value of pv at 0° C. and 760 mm.).

. • . C, = y (g- X -273 X value of pv at 0° C. and 760 mm.).
But in the last paragraph we saw that

T X 1J73 X ^v at ^° ^' anc* ^^ mm.) = 2 calories.

3
C, = -„- x 2 = 3 calories.

Thus we should expect C» to be 3 calories,
and Cp to be 5 calories.

y =| = l-667.

In the case of the gases, helium, argon, neon, mercury vapour.
which from chemical reasoning are regarded as monatonic, the
value of y found by experiment is very close to this value 1-667.

On the other hand, for gases like air, oxygen, etc., it is about
1-4, and for carbon dioxide 1-29.

If the molecules are not mere particles of infinitesimal size,
they might possess kinetic energy other than that of translation ;

THE GAS EQUATIONS 305

for example, if they were groups of atoms the group might have
energy of rotation as well as of translation. In that case when
the temperature was raised the energy of rotation would probably
increase as well as that of translation.1 We should therefore
expect C» to be 3 + m, where m represents the extra energy of
rotation ; m would increase with the complexity of the molecule.
But Cf — C. remains 2 since this is governed by the external
work.

5 + m 5

Hence y would be -5— - — , and so would be less than « ; the
u T Tn o

greater the value of m the less would be the value of y. It is
found that the more complex the molecule the smaller is the
value of 7, so that something of the kind indicated must be
going on. ~. s

It may be noted that if the molecules be not very complex, m
approximates to the value 2 for each extra atom after the first.

Thus for the monatomic gases, helium, argon, etc.

5
y = y about.

For the diatomic gases, hydrogen, nitrogen, etc.

y =-r-or 1-4 about,
o

For triatomic gases, e.g. CO2, water vapour, H2O,

9
y = -=-or 1-3 about.

This is not a strict law, however, and it breaks down at the
more complex molecules.

This is a case of the law of distribution of energy between different
d«gre«s of freedom (Maxwell).

CHAPTER XXI
THERMODYNAMICS

In the chapter on heat engines attention was drawn to the fact
that in such engines only a fraction of the heat taken in is con-
verted into useful work, and that there is always a large amount
of heat returned to the condenser or the atmosphere. As by far
the greater part of the energy needed in carrying out the various
requirements of civilization is derived from some form of heat
engine, it becomes a matter of the utmost direct practical import-
ance, apart altogether from its value in the development of the
theory of heat, to find out the conditions which determine what
fraction of the total heat supplied can be converted to work. The
present chapter is devoted to the consideration of such con-
ditions and some of the consequences which result from them.

Work done by an Engine — Indicator Diagrams. — For our
present purpose it is undesirable to confuse the issue by consider-
ing the work wasted in friction in the engine itself ; we shall
therefore deal with the work done by the steam or gas on the
piston, and not with the final output as measured by a brake.
This is done by using the relation of page 279 for the work done
by an expanding fluid ; in practice it is accomplished by using
an " indicator," an instrument which gives a continuous record
of the pressure in the cylinder for every position of the piston in
its stroke. The indicator consists essentially of a small cylinder
connected to the engine cylinder and fitted with a piston ; this
works against a spring, so that when the steam (or gas) pressure
rises the piston moves up and compresses the spring. The
motion of the piston is transmitted through levers to a pencil
which presses lightly on a sheet of paper fixed to a drum ; the
latter rotates backwards and forwards through a certain angle
corresponding with the backward and forward motion of the
engine piston.1

1 This is usually accomplished by a cord, connected through a reduo-

306

THERMODYNAMICS

307

D

The curve traced out by the indicator pencil during one com-
plete to-and-fro stroke gives a record of the conditions in the
cylinder and is called an indicator diagram. Fig. 131 represents
a diagram such as might be obtained. The ordinates represent
pressures and the abscissae which actually correspond to the
positions of the piston are proportional to the volume of the
contents of the cylinder. The work done by the steam on the
piston during the outstroke is equal (page 279) to the area under
the curve ABC, i.e. KABCL. The work done by the piston on
the steam during the instroke
is equal to the area under the
curve CD A, i.e. LCDAK. The
difference between these two
areas, i.e. the area of the closed
curve ABCDA, represents the
net work done on the piston
during one complete to-and-fro
stroke. This work, multiplied
by the number of revolutions
per minute, gives the work
done per minute, and so the
horse-power.1

The horse-power found in this way is called the Indicated

ing lever with the cross head of the piston rod, so that the movements
of the drum are a copy on a small scale of the movements of the engine
piston.

1 In actual practice the scale of pressures is known, but the precise
ratio of the abscissae to piston stroke is not. The engineer therefore
uses the diagram to determine the " mean effective pressure," i.e. the
average value of the difference between the pressure on the out stroke
and the pressure on the in stroke. This he does either by measuring
the area of the digram by a planimeter and dividing by the length
KL ; or by dividing the base into ten or twelve equal parts, erecting
perpendiculars from the middle of each part, measuring the lengths of
these perpendiculars between the top and bottom line of the diagram,
and taking the mean of these lengths. The mean effective pressure
multiplied by the area of the piston and multiplied by the length of the
stroke gives the work done ; this multiplied by the number of revolu-
tions per minute gives the work done per minute. Thus the indicated

bore"- power is given by — , where P is the mean effective pressure,

Ibs. eq. inch, L the length of the stroke in feet, A the area of the piston
in sq. inches, and N the number of revs, per minute. It will be seen
that PLA comes to the same thing asfpdv round the closed curve ; the
question is one of practical convenience.

PIG. 131.

308 HEAT

horse-power, and is written I.H.P. ; if the engine is double-acting
a similar diagram must be taken from the other end of the cylin-
der and the two values added together to get the total work
done.1

Now in all heat engines, as we saw in Chapter VIII, the idea is
to have some substance, called the working substance, which is
made to expand by heating it and so does work. In any such
operation the principle of the Conservation of Energy holds good,
so that if a quantity of heat (5H is absorbed by the substance and
external work <5W is done, the difference between W and J<5H
(where J is the mechanical equivalent of heat) must be repre-
sented by a gain in energy of the substance. For example, if we
heat some gas at constant pressure, we do a certain amount of
external work, and we also increase the kinetic energy of the
molecules, and (unless the gas is perfect) increase the internal
energy of the gas by separating the molecules against their
mutual attractions. If we call the change in the energy of the
substance (5E, we can say

J . <5H = <5W + <5E.

This statement is known as the First Law of Thermodynamics.
»fc will be seen that it is merely a particular statement of the
principle of the Conservation of Energy ; with the corollary that
heat is one form of energy.

If we perform a series of operations with the working substance
and finally bring that substance back to the same conditions
under which it started (pressure, volume, temperature, etc., the
same), then the energy of the substance is the same as before, that
is to say (3E = 0, and the total amount of heat absorbed is equi-
valent to the total amount of external work done. Such a series
of operations is spoken of as a complete cycle. It follows then
that if we can perform a complete cycle with our working sub-
stance in such a way that on the whole heat is absorbed, such
heat is genuinely converted into work, and it is clear that we can
go on converting heat to work continuously (so long as the heat
is available) by repeating the cycle.2

1 It should be noted that the effective area of the piston for the second
diagram is less than that for the first by the area of the piston rod.

a It is the necessity for providing for continuous operation that
involves the working substance being taken round a complete cycle.
In all practical heat engines this is the case: see note on page 105.

THERMODYNAMICS 309

It was shown by Carnot in a celebrated essay * that the neces-
sary condition for such conversion is that the heat should be
" let down from a high temperature to a lower one," that is to
gay heat should be supplied to the working substance when it is
at a high temperature and abstracted from it when it is at a lower
temperature ; and he showed what form of cycle would result in
the greatest possible production of work. This essay is of fun-
damental importance and is the basis of all engineering practice.
It is true that the conditions of Carnot's cycle cannot be com-
pletely realized in practice (and incidentally would require an
engine to work infinitely slowly) ; nevertheless the cycle is of
supreme importance as giving the ideal standard to be reached
and as indicating the way in which it may be approached.

The following account of Carnot's reasoning differs in one point
from the original (see footnote, page 312) ; to retain the earliest
form at this stage would only obscure the issue.

Reversible Processes — In all the operations which occur in
Carnot's cycle the process must be rever-
sible. Reversible, in the thermodynamic
sense, means that at every stage of the
process an infinitesimal change in the ex- ZE^
ternal conditions will make it go the other ^|
way. We may illustrate this by consider-
ing a simple case. Let Fig. 132 represent £^
a long metal cylinder fitted with a fric- ^Z
tionless piston and containing some water. EKI
The whole is placed in a constant-tern- zTi:
perature bath. Let p be the vapour ^ri
pressure of water at the temperature of ."_"_"_" ~~~ :
the bath, and A be the area of the piston.
If the piston be loaded so that the total

weight of it is pA, nothing will happen. But if the load be dimin-
ished ever so slightly, the water under the piston will begin to
evaporate, pushing it up, and at the same time will absorb the
necessary latent heat of evaporation from the bath. This process
can go on until all the water has turned to vapour. If at any
stage the load is increased so that it is slightly greater than pA,

1 Sadi Carnot, Reflexions sur la puissance motrice du feu et sur le»
moyens propres a la developper, 1824. An English translation by R. H.
Thureton is published by Macmillan & Co.

310 HEAT

the piston will descend, vapour will condense, and latent heat of
evaporation will be given up to the bath. Thus by changing the
load from an amount infinitesimally less than pA to one infinites! -
mally greater the whole operation has been made to go the other
way. It would be difficult in practice to carry out a corresponding
process in the case of a gas at constant temperature ; to do this
we should have to alter the load on the piston from moment to
moment so that at every stage it was infinitesimally less or
greater (as the case might be) than the force due to the gas.
Nevertheless we can conceive such a process.

A little thought will show that in both these cases the move-
ment of the piston would have to be indefinitely slow, otherwise
finite differences in pressure would arise.

Garnot's Cycle. — The working substance can be anything
(provided that it does change in volume when heated), but it is
easiest to picture it as a gas or vapour. Imagine it contained in
a cylinder, fitted with a frictionless piston. The piston and the
walls of the cylinder are made of material which is a perfect heat
insulator, but the base is of a perfect conductor. The whole can
be placed at will on either X, Y. or Z (Fig. 133). X is a source of
heat which remains at a constant temperature Ol, and can supply
any quantity of heat required. Y is a stand of some perfect heat
insulator, so that when the cylinder is placed upon it, the contents
can neither receive nor give out heat. Z remains at a constant
temperature 02» and can absorb any quantity of heat. We shall
refer to this as the sink.

Let us start with the working substance at 62 ; the volume will
depend on the mass of substance in the cylinder. The initial
state of the substance is represented by the point A on the
indicator diagram. Place the cylinder on Y and gradually
increase the load on the piston so that the substance is com-
pressed by a reversible process. As the cylinder is on Y, there
is complete heat insulation, and the process is adiabatic ; the
temperature of the working substance will rise. Let the com-
pression go on until the temperature has reached 0t. AB on the
indicator diagram represents this part.

Now place the cylinder on X, and by adjusting the load allow
the working substance to expand reversibly and isothermally.
Let this go on until the substance has reached some suitable
condition, represented by C on the indicator diagram. Transfer

THERMODYNAMICS

311

the cylinder to Y and allow the expansion to continue reversibly :
it will now be adiabatic and the temperature will fall. CD
represents this part of the operation. When the temperature of
the working substance reaches 02» place the cylinder on Z and
alter the load so that the substance is compressed reversibly and
isothermally ; DA represents this part. During the compression
heat will be given up to the sink. Continue the process until the
substance reaches the condition represented by A ; we have now

Indicator Diagram.
Fio. 133.

completed the cycle, and the conditions are the same as when
we started.1

During the cycle work is done by the substance along BC and
CD, and work is done on it along DA and AB. The net work
done by the working substance is, as already explained, equal to
the area of the indicator diagram. During the isothermal expan-

1 A is the most suitable point at which to start the cycle, since it ia
easy to know when we have got back to it. If we start at B we have
got to carry on our isothermal compression to such a point that if the
adiabatic compression takes place we shall just hit off the point B.

312 HEAT

sion the substance takes in a quantity of heat Ht from the source
X, and during isothermal compression it gives up heat H2 to the
sink Y ; the other parts of the cycle are adiabatic, so no heat
transfer occurs. Since the working substance returns to its
initial condition, the First Law of Thermodynamics tells us
that J(H! — H2) = external work done == area of indicator dia-
gram.1

It is clear also that, since every operation is reversible, we could
have gone round the cycle in the opposite direction ; in which
case we should have had to do work equal to the area of the
indicator diagram, and should have taken H2 from the sink and
put the greater amount Hj into the source. If this is done the
engine acts as a " heat pump " and the process corresponds to
the action of refrigerating machines (page 166), and also (though
less closely) to the operations in the liquefaction of gases (page
290).

Efficiency of Carnot's Process. — The function of a heat engine
being to convert heat to work, the efficiency is given by

Work done

Mechanical equivalent of the heat absorbed*
That heat is also rejected is interesting, but of no use for our
purpose — it represents heat which we wished to be converted to
work but which was not.

Work done
Thus efficiency = — r ^p — , and since we are not considering

J X -tlj

1 In his original essay, Carnot based his reasoning on the conservation
of energy, but adopted the caloric theory of heat ; he therefore assumed
that the heat given up to the sink was equal to that absorbed from the
source ; the process of dropping from the high to the low temperature
enabled it to do work. This idea of heat rather corresponds to the case
of an electric motor ; in this the current flowing out is equal to the
current flowing in and the work is done I y the electricity dropping
from the high potential to the low. The later experiments of Joule
and especially of Hirn showed that the heat rejected by an engine ia
less than that absorbed by an amount proportional to the work done.
Thus, as summarized in the First Law, heat in its own right represents
a form of energy apart from all questions of temperature. Nevertheless
Carnot's argument remains valid as to the availability of hea*. for doing
work ; it was modified by Clausius and by Lord Kelvin to accord with
the dynamical theory of heat.

It may be pointed out that Carnot implies that he does not commit
himself as to the nature of heat and adopts the caloric idea as a possible
view. Before the close of his short life in 1832 he appears to have had
a clear appreciation of heat as a form of energy.

THERMODYNAMICS 313

friction losses in the engine we can put J(H.1 — H2) for the work
done.

^

Thus efficiency — J x H Or IS. —

This is only expressing the efficiency as the fraction of the
whole heat absorbed which is converted to work.

The next proposition is to show that No engine working between
the same two temperatures can be more efficient than a reversible
engine.

Suppose we have a reversible engine working between a source
at 0!° and a sink at 02° ; let it take an amount of heat H from
the source per cycle and do work W. If we have a more efficient
engine working between the same source and sink, it means that
if it absorbs H from the source per cycle it can do more work than
W. Let this engine be arranged to drive the reversible one
backwards ; the latter then acts as a heat pump, delivering heat
H to the source per cycle and requiring work W to be done on it.
Now the efficient engine can do more than W per cycle, so that
it can drive the other and do some other work as well. But it
only takes H from the source per cycle, so that one engine returns
as much heat as the other absorbs. Thus the combined engines
take no heat from the source, but are able to do a certain amount
of extra work per cycle ; the process can go on indefinitely. By
the First Law of Thermodynamics the heat equivalent of this work
must have come from somewhere ; it must therefore have come
from the sink.1

We could, if we liked, have arranged that our more efficient
engine did an amount of work W just sufficient to drive the
reversible engine ; in that case it would absorb an amount of
heat H' from the source somewhat less than H. The combined
engines would then do no extra work, but an amount of heat
H— H' would be transferred to the source each cycle, and this
heat must have come from the sink.

Thus by a continuous process, and without external aid, w*

1 In Carnot's essay since the heat rejected by an engine was assumed
to be equal to that absorbed, it followed that no heat came from the
sink either, and this involved continuous work without absorption of
energy — in other words ' ' perpetual motion. ' ' It was therefore regarded
as a proof of impossibility. This argument does not apply when we
adopt the dynamical theory of heat, but the general conclusion remains
the same if we adopt the Second Law of Thermodynamics.

314 HEAT

should be able- either to obtain work by absorbing heat from tne
low-temperature sink, or cause heat to pass from the cold body
to the hot. As far as the First Law of Thermodynamics is con-
cerned this might be possible ; but it is contrary to all experience
that either of these things should happen. This statement is put
forward in the hypothesis which is known as the Second Law nj
Thermodynamics, and is usually expressed in one of two forms,
which really come to the same thing. "It is impossible, by
means of inanimate material agency, to derive mechanical effect
from any portion of matter by cooling it below the temperature
of the coldest of surrounding objects."

This form is Kelvin's method of expression.

Clausius puts it in the form : " It is impossible for a self-
acting machine, unaided by external agency, to convey heat from
one body to another at a higher temperature, or heat cannot 01
itself pass from a colder to a warmer body."

It is important to notice that this law applies only to con-
tinuous operations — e.g. some compressed gas when released from
pressure will do work and cool itself, but this cannot go on
indefinitely, and further the gas must originally have been com-
pressed by external agency. The Second Law is not capable of
direct proof, but we are convinced of its truth because no experi-
ment has ever succeeded in disproving it and predictions based
on it are found to be correct.

Once the Second Law is admitted it follows that no engine can
be more efficient than a reversible engine working between the
same temperatures. It follows also that all reversible engines
are equally efficient whatever the working substance ; for if one
is more efficient we can set it to drive the other backwards and
get the results of the previous paragraph. The only condition
which affects the efficiency of a reversible engine is the
temperatures at which it takes in and rejects heat ; we must
now see in what way the efficiency depends on the tempera-
ture.

It is important to remember that the results of the last para-
graph make the following reasoning quite general ; since no
engine can be more efficient than a reversible engine we can take
the Carnot cycle and any working substance which is easy to
follow and apply the results to other cases.

The working substance takes in heat during the process repre-

THERMODYNAMICS

316

V

FIG. 134.

sented by the isothermal BC (Fig. 134) ; the amount taken in is
clearly determined by conditions implied by BC, and is not
influenced by any subsequent operations. How much of this
heat is converted into work depends on the area of the indicator
diagram ; in fact the possibility of any work being done at all
during the cycle depends on DA being below BC, so that during
the return stroke the pressure on the piston at any stage is less
than the pressure on the
corresponding stage in the
outstroke. The more we
can keep down this back
pressure, the more work
will be done in the cycle,
and this means we must
keep the temperature as
low as possible. Thus the
lower we keep the tempera-
ture during the period
when heat is taken back
from the working sub-
stance * the greater will be the amount of work done in the
cycle. For example, if we return along the isothermal D'A',
which represents a higher temperature than AD (for the pressure
corresponding to any value of v is higher along D'A' than JDA),
the total work done will be represented by the area BCD 'A'
instead of BCDA.

We thus arrive at the very important result that if an engine
takes in heat at a particular temperature 6lt and rejects heat at a
temperature 02, the fraction of the heat taken in which can be con-
verted into work becomes greater as the temperature drop Ol—02 is
increased.

In practical engines, there is a limit to the lower temperature,
which cannot very well be lower than atmospheric (and is usually
higher). The only chance of making the efficiency as high as
possible is to arrange to take in heat at as high a temperature as
possible. The upper limit of temperature is decided by questions
of mechanical conditions, and to a large extent by the difficulty
of arranging for lubrication at high temperatures. It follows

1 It is clear that heat must be absorbed during some part of the
return stroke, otherwise we shall return along the adiabatio DC.

316 HEAT

also that if heat is taken in at any part of the cycle where the
temperature is below the maximum, that heat is less advantage-
ously employed ; and any heat rejected above the minimum
temperature is disadvantageous.1

Thus the best possible cycle is that of Carnot, where the heat is
all taken in at the highest temperature and rejection of heat
takes place at the lowest, the other processes being adiabatic.1

Lord Kelvin's Absolute Scale of Temperature. — The results of
the last paragraph give us a means of denning a scale of tern
perature which is independent of the properties of any particular
substance, and of obtaining a conception of the absolute zero of
temperature which has a real physical meaning. For if any
reversible engine takes in a quantity of heat H at any particular
temperature the fraction of this heat converted into work during
a complete cycle will depend only on the temperature of the sink,
and will be greater the lower this temperature is made. Now the
utmost effect we can conceive (unless we abandon the First Law
of Thermodynamics) is to convert the whole of this heat into
work ; thus if we reduce the temperature of the sink until the

1 For example, one trouble with steam engines is that some of the
steam entering the cylinder is condensed on the relatively colder walls
of the cylinder. As the pressure falls towards the end of the expansion
stroke re-evaporation takes place, so that some of the heat lost at the
beginning returns to the working substance ; nevertheless this heat,
being taken at a lower temperature, is less available for conversion to
work. It was an important point in Watts' engine that the cylinder
was jacketed with boiler steam which largely prevented this con-
densation. It will be seen that it is worse than useless to jacket the
cylinder with steam from the exhaust, which is at the lowest tempera-
ture.

2 In actual engineering practice, about the nearest approach to this
cycle is met with in the Diesel engine. This is a four-stroke oil engine,
which on the induction stroke takes in air only. This is compressed
on the compression stroke and the temperature is raised to a high
value ; oil is pumped into the cylinder in the early stages of the working
stroke and at once ignites owing to the high temperature of the air.
Thus heat is taken in by the working substance at nearly constant pressure
and slightly rising temperature during the first part of the stroke ; the
oil is then cut off and the contents of the cylinder expand for the rest
of the stroke and are then let out by the exhaust valve during the
fourth stroke. The thermal efficiency of this engine, judged by its
indicated horse-power, is about '4, which is higher than other oil engines
and something like twice that of the best steam engines. The mechani-
cal efficiency is rather low, so that the thermal efficiency judged from
the B. H.P is about -32, which is much the same as the best oil engines,
but considerably higher than steam.

THERMODYNAMICS 317

whole of the heat taken in is converted to work we have got to
the lowest conceivable temperature — an absolute zero in the
strictest sense. (It must be again emphasized that we are dealing
throughout with complete cyclical operations, so that no work is
got from the intrinsic energy of the working substance.) If the
temperature of the sink is higher than this absolute zero, the
fraction of the heat converted into work, or in other words the
difference between the heat absorbed and the heat rejected will
depend simply on temperature of the source and sink and on
nothing else. On the absolute thermodynamic scale tempera-
tures are therefore defined as follows : 1 if a reversible engine
takes in a quantity of heat Hx at one temperature 0X and rejects
a quantity H2 at the lower temperature 0a,

II=!L
02 H2

and in the particular case we have just considered, when H2 = 0,
02 also = 0.

The scale of temperature denned in this way is called the
absolute thermodynamic scale and is due to Lord Kelvin.1

The thermodynamic scale is of immense importance, since for
the first time we have obtained a scale which does not depend on
the properties of any arbitrary substance. We must now see
how it is related to scales of temperature defined in other ways.
For this purpose we will compare it with the perfect gas scale by
considering the indicator diagram when a perfect gas is taken

1 See note at the end of the chapter.

2 The idea of obtaining an absolute scale of temperature by making
use of Carnot's principle was first put forward by Lord Kelvin (William
Thomson as he then was) in a paper in Proc. Camb. Phil. Soc., reprinted
in the Phil. Mag., 1848. In this first paper he adopted the notion ol
Carnot's essay of 1824 that heat did not disappear and that work was
done by the passage of heat from high to low temperature without
change in the quantity. Indeed he says " the conversion of heat (or
caloric) into mechanical effect is probably impossible," but adds a
footnote : " This opinion seems to be nearly universally held by those
who have written on the subject. A contrary opinion has been advo-
cated by Mr. Joule of Manchester." Later on he modified his views,
and his clear conception of the universal importance of the doctrine
of the conservation of energy and the development of its consequences
is perhaps the greatest of his contributions to the advancement of
knowledge. This may be seen in his collected papers. The history ol
the absolute scale is of interest as showing how the Second Law ol
Thermodynamics is an entirely distinct proposition and does not in any
way follow from the First.

318

HEAT

round a Carnot cycle. It must be remembered that a per-
fect gas satisfies the following conditions : —

(1) No internal work is done when the volume changes.

(2) The gas obeys Boyle's Law exactly at all temperatures.

Temperature on the per-
fect gas scale is defined by
the relation pv = RT where
R is a constant and T the
temperature defined in this
way.

Let ABCD (Fig. 135) re-
present the indicator dia-
gram for the perfect gas
taken round a Carnot cycle.
AB and CD are adiabatics
and BC and DA isother-
mals. Let the tempera-
ture of BC defined on the gas scale be T! and defined on
the thermodynamic scale be 8^ Let AD be T2 or 62. Let H, be
the heat taken in along BC and H2 the heat rejected along DA.
Now since no internal work is required to separate the molecules
of the gas, and there is no rise in temperature alonp BC, we have
J x H! = external work done by the gas in passing from B to C.
Similarly

J x H2 = external work done on the gas in passing from D to A.
thus J X H! =fpdv along BC.

V

FIG. 135.

But for the isothermal BC we have pv

«„,.?

,-. J x

-dv

Similarly J X H2 - RT2(logev8-logev4) = RT2loge^-

(1)

THERMODYNAMICS 319

Now to find the relation between vlt v2, v,, v4 we have from the
adiabatica

...... (2)

and from the isothennals we have
Pivi =

.

If we multiply equations (2) and (3) we shall eliminate the p'a
and obtain

er - cr

t>a

Vo V«

Thus in equation (1) we have log, — = log, —

and therefore vj^ = pm* or W" = «r"
Jtl2 ±vl2 ±±2 A2

But 0j and 0a are defined by the relation

Therefore £- = ^r

and since the isothermals were selected quite arbitrarily we have
the proposition of fundamental importance that The thermody-
namic scale coincides with the perfect gas scale.

Now Callendar and others l have shown how to calculate the
value of a temperature measured on a hydrogen or nitrogen
thermometer in terms of the perfect gas scale from a knowledge
of the deviation from Boyle's Law and the porous plug experi-
ment.

Thus it is possible actually to measure any particular tempera-
ture and then obtain its value on the thermodynamic scale which
is independent of the properties of any substance and depends
only on the laws of energy.

It may be pointed out that since the deviations of the hydrogen
and nitrogen scales from the perfect gas scale are very small,
and practical temperature measurements up to 1,600° C. are
1 Cp. Preston's Theory of Heat, 3rd edition.

320 HEAT

based on one or other of these two scales, it is sufficient in engi-
neering practice to say that the limit for the efficiency of any
engine is given by the relation

Ht-H2 TX-T2

Hi T!

where the temperature is measured in the ordinary way, but, of
course, expressed as degrees Absolute, not degrees Centigrade.

Note on the Definition of Temperature on the Thermo -
dynamic Scale. — In the account of the thermodynamic scale it was

fi TT

stated that temperatures were defined by the relation -^ = ^ ; it is

c/2 ±±2

convenient to give in a more concise form the detailed reasoning
which shows that the definition of temperature in this way gives
a scale which is perfectly consistent. The following treatment
appears to the writer to be the clearest, and has been used by
him for a good many years.

Let a reversible heat engine work between two temperatures
Ol and 02 defined in any arbitrary way.

TT _ TT

The efficiency of the engine, which is given by -~r — - (see

page 313) depends only on 6^ and 02 ; we can express this by the
relation

TT _ TT

^ - = /(0i02) where /(0A) is some function of 0! and 02

whose form is determined by the way in which the temperatures
are defined.

We can write this relation in the form

TT

~= <p(0i02) where <p(0102) is some other function of 0! and 02

JJ-2

related to /(#i02) by the condition that since

2 '

Let a second reversible engine work between 02 and 03, and let
it be of such a size that it takes in H.2 at 02 per cycle.
Then as before the efficiency = /(0208),

TT

and -vf- = Q?(0203).

-tt-a

If now the engines are arranged to work as a single engine and
the strokes are timed so that as the first rejects its heat H2 the
second takes in the H2, we have a single reversible engine in

THERMODYNAMICS 321

which the working substance happens to be contained in two
cylinders ; this engine takes in heat II j at Ql and rejects H, at
0, and therefore

BV.C by ordinary multiplication

H! H2 Ht
H * H — H
9>(0i02) x ?(0203) = 9>(0A).

If then 02 disappears when <p(0!02) is multiplied by ^(0203) it
means that the form of the function must be a fraction in which
some function of 0j occurs in the numerator and the same
function of 02 in the denominator, or

.HI _

H2 ~ V(d2y

Thus, however we define 0, we can get a consistent scale of
temperature provided that we define it in such a way that
H! some function of 0!
H2 ~~ the same function of 02«
The function y>(0) can be of any form we choose.

The simplest way is to define 0 so that y(0) is simply 0 ; we
arrive at a consistent scale by defining it in such a way that

CHAPTER XXII

TRANSFORMATION OF ENERGY—DEGRADATION
AND DISSIPATION OF ENERGY— ENTROPY

Conservation of Energy and the First Law of Thermodyna-
mics.— During the latter half of the nineteenth century, the
researches of Joule, Him and others, and particularly the work
of Lord Kelvin, placed on a firm foundation the exceedingly
important Principle of the Conservation of Energy. This prin-
ciple may be stated in the form that there is some identity
underlying the phenomena of mechanics, heat, light, electricity,
etc. These phenomena are different manifestations of some-
thing which we call energy. The different forms are mutually
convertible, that is to say if we have the power to produce certain
phenomena in light and this in some way disappears, there is left
in place of it the power to produce certain phenomena in heat or
electricity or mechanics, as the case may be. Moreover there is
a definite numerical relation between the capacity to produce the
optical phenomena measured in suitable units, and the capacity
to produce the phenomena in heat which appears in place of it.
Energy cannot be created or destroyed; if one form of it dis-
appears a corresponding amount in another form appears. This
general principle is the Principle of the Conservation of Energy ;
the First Law of Thermodynamics is a particular part of it which
says that heat as measured in ordinary thermal units is one of
the forms of Energy. Put in its simplest form, the First Law of
Thermodynamics is given by the relation J x H = W where H
is a quantity of heat, W the work done when H disappears
(and no other form of energy comes into the question) or the
quantity of work which disappears when H is produced, and
J a constant depending on the units chosen to measure H
and W.

Since however th3re might be an appearance or disappearance
of other forms of energy besides mechanical work, the moat

322

TRANSFORMATION OF ENERGY 323

general statement of the principle is that of page 308, namely
J x <5H = <5W -f <5E.

Distinction between Heat Energy and other Forms. The
Second Law of Thermodynamics . — Although the different forms
of energy are equivalent in the sense that they can all be ex-
pressed in terms of a common standard, and that if a certain
quantity of one form disappears a corresponding quantity of
another form appears somewhere, there is an important dis-
tinction between heat energy and other forms. We can carry
out, or conceive of, processes by which one of these latter forms
can be completely converted into another, or any form can be
converted into heat ; but there is a limitation on our powers of
converting heat into other forms.

It has been shown in the last chapter that heat can only be
converted to mechanical work when it is let down from a body
at high temperature to one at a lower temperature, and that even
when the circumstances are most favourable, i.e. all the opera-
tions are reversible in the thermodynamic sense, the whole of the
heat is not converted to work unless the sink is at the absolute
zero of temperature. If then we have an isolated system possess-
ing heat energy, and all parts of the system are at the same
temperature, none of the heat energy can be converted to mechan-
ical energy. It is thus apparent that there is a distinction
between heat energy and other forms in that we can never
convert the whole quantity of heat into work, and in some cases
we cannot convert any of it.

It is conceivable that this distinction is due to our own limita-
tions rather than to a fundamental difference in the nature of the
energy. If we regard heat energy as the kinetic energy of the
ultimate particles of matter the limitation means that we are
unable to isolate the particles and direct their kinetic energy so
that it may be available for our own particular purpose. Whether
this is so or not, we must regard heat energy as a lower grade than
other forms in that we have not a free choice in the use of such
energy. It should be noted that this proposition is in no way a
consequence of the Principle of the Conservation of Energy ;
indeed the first recognition of it was founded on a train of reason-
ing which involved a violation of that principle. The proposition
is expressed in the Second Law of Thermodynamics which is to be
found stated in many different forms, but which practically comes

324 HEAT

to a quantitative statement of the distinction between heat
energy and other forms.

Degradation and Dissipation of Energy. — If we have a system
of which some parts are at a higher temperature than others we
can convert some of the heat energy into work by using a revers-
ible engine which takes in heat from the hot parts and rejects a
portion of it to the cold. On the other hand, if heat passes
directly, e.g. by conduction, from the hot parts to the cold, such
heat is no longer available for conversion to work. In this case
the system is said to have suffered a degradation of energy since
its energy is now less available. There is a constant tendency
for heat to pass from places at high temperature to those at lower
temperatures, thus bringing about this degradation of energy.
Again in any actual conversion of energy from one form to
another there is always a certain amount wasted in such ways as
friction, turbulent motion of fluids, electrical resistance, etc., the
wasted energy being converted to heat. This conversion by the
way of a portion of the energy into heat is spoken of as the
dissipation of energy.

Thus although the total amount of energy in an isolated system
remains constant, and any changes which occur merely alter the
form of the energy and not its amount, yet in any such change
there will be a certain amount of dissipation of energy into heat.
And whether changes occur or not there is a tendency for heat to
pass from places at high temperature to those of lower tempera-
ture; there is no known method of complete heat insulation.
Thus in addition to the Principle of the Conservation of Energy
we have the very important practical Principle of the degradation
and dissipation of energy which tells us that the energy of a
system is always tending to become less available for general use.

Entropy. — We must now deal with a conception which is of
great use to the engineer in calculations of the behaviour of heat
engines and which has many applications in physical problems.
We have seen that Carnot, in his essay of 1824, regarded his
engine as rejecting the same amount of heat as it took in, the
work being done by the heat dropping from a high temperature
to a low : this conception is rather like the case of a water motor
where the quantity of water which comes out of the waste pipe
is equal to that which flows in, work being done because the pres-
sure of the water on the inlet side is greater than that on the

TRANSFORMATION OF ENERGY 325

outlet. It is now established that this is not the case with a heat
engine ; some heat disappears, and there is a conversion of heat
to work. The conception of the physical quantity known as
entropy does provide us with something which under certain
conditions may be regarded as passing through the heat engine
just as the water passes through the water motor or the electric
current through an electro -motor without altering in amount.
We will first define entropy and then consider its meaning.

Definition. — If a substance at a temperature (absolute) 0
absorbs a quantity of heat H it is said to receive an amount of

TT

entropy 5-. If the temperature does not remain constant during

the process we must divide the quantity of heat into small
elements (5H such that during the absorption of each element the
temperature remains practically constant and regard the entropy

received as

C dH.
J «'

Consider the system made up of a source at 0lt a sink at 02, and
a Carnot engine working between them. During the outstroke,
the source gives up an amount of heat H^ to the working sub-

TT

stance. It therefore gives out an amount of entropy z~^, and

H,

the working substance receives entropy — .'

0i

While the working substance is expanding adiabatically it
neither absorbs nor rejects heat, and so it neither gives nor receives
entropy. On the instroke the working substance rejects H2 at 02,

XT

and so it gives out entropy by -^ ; at the same time the sink

TT

takes in H2 at 02 and so receives entropy -^. During adia-

batic compression there is again no change of entropy.

H H

But in a Carnot cycle -^l = «-2. Hence the entropy lost

°l °2

by the source is equal to that gained by the sink, and the
entropy taken in by the working substance is equal to that
which it rejects. Thus for the system as a whole there is

1 Since the operation is reversible the temperature of the working
substance while it is taking in heat is indefinitely near that of the source
so that 0, refers to both of them.

326 HEAT

no change of entropy j a certain amount has been transferred
from the source to the sink, passing through the engine on the
way, just as a current passes from the high potential side to
the low potential through an electro -mot or.

The Entropy of a System tends to increase. — The analogy
between entropy and quantities such as quantity of electricity
must not be pushed too far ; entropy has the distinctive property
that it tends to increase. For suppose we have some part of a
system at a temperature Olt and a quantity of heat (5H passes
directly from it to a part at a lower temperature 62. The hot

(5H <5H;

part loses entropy -«— and the cold part receives entropy -5—
"i "2

the cold part therefore receives more entropy than the hot part
loses, so that the entropy of the system is increased. Again, if
there is any dissipation of energy through friction, this energy is

TT

converted to heat and there is a gain of entropy -TJ- where 0

is the temperature of the part receiving the heat, and no corre-
sponding loss of entropy in other parts. Both of these effects tend
to happen, and so there is a tendency for the entropy to increase.
We have just seen that if some of the heat energy is converted
to work by a reversible process the entropy of the system does

TT TT

not change, for -^ = ~. By no process can we make

H-2 . H-i

-0— less than -Q-, for if we did we should have converted a greater

fraction of Hj into work than a reversible engine can do. Thus
for any isolated system the only possible changes are those which
result in the entropy either remaining the same or increasing. It
follows also that if the entropy increases some of the energy
which might have been available for doing useful work has been
degraded into the less available heat.

Calculation of the Entropy of a Substance. — Although the
change in the entropy of a substance which is receiving heat from
a source is not necessarily the same as the change in the entropy
of the source (unless it receives its heat reversibly, in which case
the temperature of the substance is sensibly equal to that of the
source throughout the process), it is quite possible to calculate
this change in many cases. The first point to establish is that
the entropy oJ a substance in any particular state depends only

ENTROPY

327

on that state and not on the way it arrived there. To do this we
must show that when a substance is taken round a complete cycle

j -£— = 0. To begin with we will show how to take it round any

reversible cycle.

It should be observed that to make the cycle reversible is only
a question of adjusting the external conditions. Let the thick

V

Fio. 13ft.

line represent any complete cycle. Draw a number of adiabatic
curves represented by dotted lines, and let ab, cd, ef represent
portions of isothermal curves corresponding to temperatures 8lt
0t, 05, etc., and gh, kl, mn portions of isothermal curves corre-
sponding to temperatures 02» #4» #6» e*°- ^ we are provided with
a number of sources at Blt 03, 05, etc., and sinks at 02, 04, 06» etc.,
we can take the substance reversibly out along the zigzag path
abcdef . . . and home by ... Ikhga.

If dHlt (5H3, etc., represent the heat absorbed along ab, cd, e),
etc., and (5H2, <5H4 the heat rejected along hg, Ik, etc., then since
ab and hg are the isothermals of the Carnot diagram abhy
6Ht <5H2

Similarly, since cd and Ik are the isothermals of a Carnot
diagram cdlk

328

HEAT

CTTT

~7p~ = — fl~* and so on< Thus for the complete zigzag path we have

<5H3

3

5

UJ-J-4 wxA2 A

" =0

By increasing the number of sources and sinks, and so taking
the isothermals and adiabatics closer together, we can in the
limit make the zigzag path practically coincide with the thick
line, in which case we shall have taken the substance reversibly
round the path and equation (1) becomes

— — = 0, where the sign (/) means the value of the in-

tegral taken round the complete cycle.

Let A and B (Fig. 137) represent any two states of the sub-

(J)

n

V

FIG. 137.

stance, and let I and II represent any two possible ways of getting
the substance from state A to state B. It will be assumed that it
is possible to adjust the external conditions so as to make either of

these paths reversible ; the value of I — for any portion is

determined by the path.1

Taking the closed cycle from A up I and down II we have

({)?-«-

1 It must be remembered that 0 is the temperature of the substance
itself at any point. The amount of heat taken in and the temperature
of the substance is fixed by the nature of the path. It is necessary to
assume that the temperature of the substance is always uniform

ENTROPY 329

A

But since the path II can be made reversible
A B

•> (I?) -(?)-«

NA XI XA 7II

B

f /7TT
Thus I -0- does not depend on the path.

A

That is, change in entropy in passing from A to B depends onb

on the states A and B and not on the way in which B is reached

If then we knew the value of the entropy of a given mass of a

substance at any state we can calculate its value at any other

f dR
state provided we can evaluate the integral — along any

path from one state to the other. We do not, however, know
the absolute value of the entropy corresponding to a substance in
any particular state, but this does not matter very much in actual
practice since we are usually concerned only with changes in
entropy. In engineering work it is usual arbitrarily to fix some
particular state as the starting point and call the entropy at
that point 0. We can then give a value for the entropy at other
states.1

Thus, in the case of steam, the starting point is taken as water
at 0° C. ; tables have been constructed showing the entropy of
unit mass of steam at various temperatures and pressures.1

It will perhaps facilitate understanding if we work out the

1 It should be observed that since we do not know the absolute value
of the entropy we cannot compare the entropy of one substance with
that of another.

2 Callendar's Abridged Steam Tables — Edward Arnold & Co.

330 HEAT

entropy of a pound of saturated steam at 100° C., leaving out
certain corrections which rather complicate the issue.

Starting with water at 0°C. or 273° A., we can get to steam at
100° C. or 373° A. by any convenient route. Let us imagine we
heat the water from 0° C. to 100° C. under sufficient pressure to
prevent evaporation, and then let it evaporate at constant
pressure of 760 mm. In heating the water we shall not make a
serious error if we regard its specific heat as constant (cp. page 128,
and equal to 1. To raise the temperature of 1 Ib. of water through
6T° will therefore require 1 x 6T or <5T Thermal Units.

The change in entropy in passing from 273° A. to 373° A. will
therefore be

373

f

dT

— - = log. 373 -log, 273.

T
273

In order to evaporate the water at 100° C. or 373° A. under 760
mm. pressure we must supply it with the latent heat of evapora-
tion at this temperature, which we may take as 539 T.U.
The gain in entropy during this operation will be

539
373*
The entropy of the saturated steam will therefore be

539
log. 373 - log. 273 + ^

539
= 2-3026(loglo 373-loglo273) + ^

= -3120 + 1-445

= 1-7570

The more accurately worked tables of Callendar give 1-7573.
Entropy is usually denoted by the symbol 9?.

Applications of the Idea of Entropy. — The change in entropy
of a substance when passing from one state to another is, as we
have stated, simply determined by the two states. The amount
of heat taken in or rejected depends not only on the initial and
final states, but also on the path by which the change takes place.
Tt will be obvious that since the entropy change g?2 ~~ ?*i ^s defined

f dH.
as -£ — the amount of heat absorbed or rejected will be smaller

ENTROPY 331

the lower the average temperature during the change. If we
know in some way the values of the entropy, and also the
temperature at all parts, we can at once get the amount of heat
absorbed. Take, for example, the case of the working substance
of an engine taking in heat at a constant temperature 0X. If q>!
and q>2 are the initial and final values of the entropy of the sub-
stance, and the heat is taken in reversibly so that 0t represents
the temperature of the substance as well as that of the source,
Heat taken in = 0i(<?>2— 9i)'

In a practical case it is probable that the heat would not be
taken in reversibly, for the temperature of the working substance
at some parts of the path would fall below 0j (in fact it would
have to do so if the heat passed into it at a finite rate), and so even
though the substance started and finished at 0j the average
temperature along the path would be somewhat less.

But as the substance finishes at the state where the entropy is

<p2, it follows that the heat taken in is I d'd<p where 0' represents

to

the temperature of the substance itself at each stage. In general
we cannot find the various values of 0' ; all we know is that
on the average it is less than 0t.

Thus the heat taken in is less than &i(<p2— 9>i), where 04 is as
before the temperature of the source supplying the heat.

We have in fact the general relation for the heat absorbed
<5H = 8d(p for reversible actions
<5H < 68(p for irreversible actions,

where 0 in each case represents the temperature at which the heat is
supplied ; the second relation is brought about because the
temperature of the substance in general differs by an unknown
amount from that of the source, dtp in each case is the change in
entropy of the substance.

Temperature Entropy Diagrams. — Much information as to
the conditions in a heat engine can be obtained by means of a
diagram of the changes in the working substance in which the
co-ordinates are temperature and entropy ; such diagrams are
often referred to by engineers as 0<p diagrams. We can illustrate
this by drawing such a diagram for a Carnot cycle (Fig. 138).

Let the working substance take in heat at Ov This isothermal

332

HEAT

expansion is represented by AB. During the process the entropy
of the substance increases from cp^ at A to q>2 at B. During an

adiabatic change (5H = 0, and so -^- = 0, so that an adia-

J 0

batic represents a line of constant entropy or an " isentropic.'
Let BC represent the adiabatic expansion. CD represents the
isothermal compression, during which the entropy changes back
from 9?2 to q>lt The substance then returns to A by the adiabatic
DA, during which there is no change in entropy. The heat taken
in along AB is O^y-, — g^), and that rejected along CD is 02(9>2 — <Pi)

^ Entropy <f&

FIG. 138.

The amount converted into work is therefore (6l— 62)(<p2 —<PI),
that is to say the area of the diagram ABCD.

In actual engines the cycles in use differ from a Carnot cycle
in various ways owing to practical considerations ; if we can draw
the Oq> diagram for the cycle, the area of the diagram will repre-
sent the heat which is converted to work. This statement is not
of course a fresh law ; it follows from the definition of entropy

as

frfH
J «'

Such cycles usually have some portion which is irreversible :
nevertheless we can still determine the conditions of the path in
some cases and so draw an entropy temperature diagram in which
the temperatures really represent the temperature of the sub-
stance, so that (5H = Qd<p.

If however there occur also fortuitous irreversible changes, e.g.
conduction to the cylinder walls, the temperature of the working

ENTROPY 333

substance will differ by an indeterminate amount from that which
is credited to it on the diagram. These fortuitous changes mean
that less heat is converted to work than the area of the 0<p dia-
gram. 1

Thus the 6g> diagram tells us the amount of heat which ought
co be converted to work and enables us to decide how far one
cycle is better than another apart from fortuitous losses. The
indicator diagram shows how much work is being done on the
piston, and so we can see whether these fortuitous losses are
occurring.

It may not be out of place to emphasize the fact that all this
conception of entropy is merely a way of expressing in a more
easily visualized form the consequences of the Second Law of
Thermodynamics.

It has been developed very much further than is here indicated,
and a large number of equations formed which represent the
possible behaviour of systems under a variety of conditions.
One such equation is a good deal used in Chemistry under the
title of the Phase Rule. All these equations are merely develop-
ments of the First and Second Laws of Thermodynamics ; their
treatment is more suited to a larger work than the present.

1 5H ^ 9d(f>, but is not greater. This comes in in two ways. SH is
the heat absorbed, so that heat rejected is negative. Thus if — 5H^ — 65$
the heat rejected is numerically greater than 6 x decrease in entropy.
Thus less heat is absorbed and more rejected. It is easily seen that if
the working substance is giving out heat to a sink it is likely to be at a
higher temperature than the sink.

QUESTIONS
PART I

CHAPTER H

1. Describe a method of filling and sealing a thermometer.

2. Why is the bulb of a thermometer usually made cylindrical ?

3. Explain what is meant by the " fixed points " on a thermometer.
Describe how you would determine them for a given instrument,
pointing out the reasons for any precautions you would take.

4. In using a hypsometer to determine the upper fixed point of a
thermometer, why is it necessary to have a manometer and also to
read the barometer ?

5. What do you understand by a scale of temperature ?

6. Express the following temperatures on the Centigrade scale:
98-4° F., 350° F., 10° F., - 10° F , 20° R.

7. Express the following temperatures on the Fahrenheit scale :
15° C., - 15° C., - 273° C., 10° R.

8. Describe the general arrangement of a mercury thermometer
suitable for temperatures up to 400° C.

9. Describe some form of maximum and minimum thermometer.

10. Explain as clearly as you can what is meant by the temperature
of a body : in the course of your answer distinguish between tempera-
ture and heat, and point out the fundamental test which decides
whether two bodies are, or are not, at the same temperature.

11. Explain how a thermometer is graduated, what is the precise
meaning of 50° C. on the mercury -in-glass scale ?

CHAPTER III

1. What is meant by the coefficient of linear expansion of a sub-
stance ?

The iron tubular bridge over the Menai Straits is rather more than
1,500 feet long. What would be its increase in length for a change
of temperature from 0° C. to 18° C. ? (Coefficient of linear expansion
of iron = -000012.)

2. A brass metre scale is correct at 0° C. : what would be its length
at 20° C. (Coefficient of linear expansion of brass = -000019.)

3. The distance between two points was measured by means of the
above scale and appeared to be 40-5 cm. If the temperature was 15°
C., what was the true distance between the points ?

334

APPENDIX 335

4. A steel measuring tape, 66 feet long, is correct at 60° F. : how
much would it contract if cooled to 32° F. ? (Coefficient of linear
expansion of steel = -000011 per degree Centigrade.)

5. Certain types of vernier callipers are made reading to 1/50 mm. ;
they are constructed of steel. If such an instrument were used to
measure objects up to 10 cm. long in a laboratory where the tem-
perature might vary through a range 20° C., would it be worth while
correcting the reading for temperature ?

6. Describe a method of determining the coefficient of linear expan-
sion of a metal such as copper or iron.

7. Describe an accurate method of determining the coefficient of
linear expansion of a solid ; and discuss the degree of accuracy to
which the method may be trusted.

8. A cylindrical bar of iron, £ inch in diameter, just fits at 15° C.
with its ends in contact with two fixed stops. If the bar is heated
to 200° C. and the stops cannot yield, what is the fo'rce exerted on
them ? (Coefficient of linear expansion of iron = -000012 ; Young's
modulus, compression or tension, = 12,500 tons per sq. inch.)

9. During a demonstration of rapid fire at Hythe with a rifle
(S.M.L.E.) in which forty-two rounds were fired in a minute, it was
noted that at the start the end of the barrel was flush with the nose-
cap, and at the end it was projecting quite £ inch. Calculate the
average temperature of the barrel (assuming it started at 10° C.).
Initial length of barrel about 2 feet 1 inch ; coefficient of linear expan-
sion of steel = -000011.

10. The diameter of a brass cylinder, as measured by a pair of steel
callipers, is 3-25 cm. when the temperature is 22° C. Find (a) its true
diameter at 22° C., (6) its diameter at 0° C. The callipers are correct
at 0°C.

CHAPTER IV

1. In practical life, the thermal expansion of solids can sometimes
be made to serve a useful purpose, while in other cases it is a nuisance
which has to be overcome by some compensating device. Give an
example of each case.

2. Explain clearly why any attempt to boil water in a thick glass
vessel is almost certain to crack the vessel, while in the case of thin
glass the operation can be carried out quite safely. Illustrate your
answer by considering the behaviour of vessels made of fused silica.

3. What is the object of annealing glass vessels and glass joints
after they are made ? Describe any experiment which illustrates
the necessity of this process.

4. Describe and explain the action of any form of thermostat.

6. Describe the construction and working (a) of a compensated
balance wheel, (6) some form of compensated pendulum.

6. When it is necessary to seal a wire through glass, platinum ia
generally used. Why is this ?

336 APPENDIX

7. Nickel steel alloys have certain important properties. What
is the special practical importance (a) of the alloy with 36 per cent,
of nickel, (b) that with 45 per cent, of nickel ?

8. Describe the construction of an invar pendulum. Why are
balance wheels of watches not made of invar ?

CHAPTER V

1. Distinguish between absolute and apparent expansion of a liquid.
How would you determine the coefficient of apparent expansion of a
liquid such as mercury ?

2. A glass pyknometer containing 250 gm. of mercury at 0° C. is
heated to 100° C., and when reweighed is found to contain 246-09
gm. Calculate the coefficient of apparent expansion of mercury.

3. A specific-gravity flask when empty weighs 13-45 gm. It is
filled with alcohol at 15° C. and found to weigh 72-36 gm. It is then
placed in a water-bath and maintained at 50° C. for some time. When
reweighed the flask and its contents are found to weigh 70-27 gm.
Calculate the coefficient of apparent expansion of alcohol.

4. In experiments with pyknometers or weight thermometers the
value of the coefficient is worked out by comparing the weight of
liquid expelled with that of the liquid remaining in the vessel. Explain
why this is so, and point out the error which would arise if the weight
of liquid originally in the vessel were taken instead of the weight of
liquid finally left in.

5. The coefficient of absolute expansion of mercury has been deter-
mined with great care, and the results can be used to standardize
glass pyknometers. Calculate the coefficient of cubical expansion of
the pyknometer in Question 2, taking the coefficient of absolute expan-
sion of mercury between 0° and 100° C. as -0001825.

6. What is the relation between the coefficient of linear expansion
of a solid and its coefficient of cubical expansion ? If the coefficient
of linear expansion of fused silica is -0000005, what is the coefficient
of cubical expansion of a vessel made of fused silica ?

7. For nearly all purposes, the coefficient of expansion of a liquid,
as determined by means of a fused silica pyknometer, may be regarded
as the absolute expansion of the liquid. Illustrate the truth of this
statement by considering the error introduced by neglecting the
expansion of the silica in the case of (a) mercury, (b) glycerine, of which
the coefficients of absolute expansion are respectively -0001825 and
-00053.

8. Describe with the aid of a graph the changes in volume which
take place when water is gradually heated from 0° C. to about 15° C.
How would you illustrate these changes experimentally ?

9. Explain clearly how a knowledge of the ratio of the densities of
a liquid at two different temperatures enables the coefficient of expan-
sion to be calculated.

APPENDIX 337

10. A glass bulb (loaded with shot inside) is weighed in air, and found
to weigh 49-360 gm. It is then suspended in alcohol so as to be
completely immersed. When the alcohol is afc 15° C. the apparent
weight of the bulb is 10-237 gm. ; with the alcohol at 30° C. the
apparent weight is 10-860 gm.

Calculate the coefficient of apparent expansion of alcohol.

11. Describe a method by which the coefficient of absolute expansion
of mercury has been determined. In your description explain how it
is that the expansion of the tube does not come into the result.

12. Regnault's apparatus for determining the expansion of mercury
was a great improvement on that of Dulong and Petit. What are the
chief points in which the improvement consists ?

13. In an experiment made with Dulong and Pe tit's apparatus to
determine the expansion of mercury, the column at 0° C. was 65-2 cm.
high and the column at 100° C. was 1-18 cm. higher. Calculate the
coefficient of absolute expansion of mercury.

14. In the experiment to Question 13, to what degree of accuracy
should the difference in height of the two columns be measured in
order that the uncertainty in this measurement may not produce a
greater probable error in the result than would be produced by measur-
ing the coefficient with a silica pyknometer and neglecting the expansion
of the vessel ? (For expansion of silica, see Question 6.)

15. The pressure of the atmosphere is found from a barometer
reading. The height of column of mercury is read off on a brass scale
(correct at 0° C.) and found to be 75-51 cm., the temperature being
16° C.

Calculate the true pressure, expressed as a column of mercury at
0°C.

(Coefficient of linear expansion of brass, -000019 ; coefficient of
absolute expansion of mercury, -000182.)

CHAPTER VI

1. State Boyle's Law, and describe an experiment by which it may
be verified.

2. Gases such as hydrogen and oxygen are often stored under pres-
sure in steel cylinders ; the usual pressure in such cylinders when
" full " is 120 atmospheres. What would be the capacity of a cylinder
intended to hold 20 cubic feet of oxygen (measured at atmospheric
pressure) ?

3. On fitting a pressure gauge to a 10-cubic-foot oxygen cylinder
(which had already been used) the pressure was found to be 80 atmo-
spheres. How much oxygen was there still available ?

4. In filling a barometer tube, a certain amount of air was acci-
dentally admitted. It was found that when a standard barometer
read 76-5 cm. the faulty barometer stood at 75-4 cm. and the surface

Z

338 APPENDIX

of the mercury was 5-8 cm. from the closed end of the tube. On o
later occasion the faulty barometer read 74-5 cm. : what would have
been the reading of a correct one ?

6. If the temperature of a given quantity of gas changes and the
pressure is kept constant, what is the relation between the volume
of the gas and its temperature ?

Describe an experiment for determining the coefficient of expansion,
at constant pressure, of a gas such as air.

6. Describe a method of determining the pressure coefficient for a
gas at constant volume.

Why would you expect the pressure coefficient of a gas to be equal
to the volume coefficient ?

7. The pressure of the air in the tyres of a car was 75 Ib. per square
inch when the temperature was 15° C. Later on the car stood in the
sunshine and the temperature went up to 25° C. Calculate the
pressure of the air in the tyres, assuming that the latter did not
expand.

8. In a determination of the pressure coefficient with an apparatus
of the type of Fig. 37, it was found that with the bulb in melting ice
the gas was brought to its fixed volume when the level of the mercury
in the open limb was 3-2 cm. below that in the other. With the bulb
in steam, the mercury in the open limb had to be 23-2 cm. above the
other. The barometer reading was 75-5 cm. Calculate the pressure
coefficient. (Temperature of steam at 75-5 cm. pressure = 99-8° C.)

9. Describe a form of gas thermometer.

Gas thermometers are usually of the constant -volume type. Can
you suggest any reason for this ?

10. Gas thermometers have certain great advantages, but also have
disadvantages which make them unsuitable for use on many occasions.
Mention some of the chief advantages and disadvantages. What is
the principal use of a gas thermometer ?

11. For a certain constant- volume hydrogen thermometer, the
pressure necessary to bring the gas to its fixed volume when the bulb
was in melting ice was 1,000 mm. of mercury : with the bulb in steam
(at standard pressure) the pressure required was 1,365 mm. In the
course of a test of a mercury thermometer, the latter was placed along-
side the bulb of the gas thermometer in a water-bath : the pressure
required to bring the hydrogen to its fixed volume was 1,212 mm.,
while the mercury thermometer read 60° C. What was the true
temperature of the bath (as given by the hydrogen thermometer) ?

12. What is meant by the absolute scale of temperature ?
Explain how the use of this scale simplifies calculations of the volumes

of gases at different temperatures.

13. What is meant by the absolute zero of temperature ?

What is its value (a) on the Centigrade scale, (6) on the Fahrenheit
scale ?

14. A certain quantity of gas was formed to occupy a volume of
450 c.c. when its temperature was 16° C. and its pressure 75 cm. of
mercury. What volume would it occupy at 0° C. and 76 cm. of
mercury ?

APPENDIX 339

15. A cubic foot of hydrogen, measured at 0° C. and 760 mm. of
mercury, weighs -0056 Ib. What would be the weight of a cubic foot
of hydrogen measured at 15° and 760 mm. ?

16. In calculations on the buoyancy of airships it is necessary to
know the density of the air at different elevations. If a cubic metre
of air at ground level, when the temperature is 9° C. and the pressure
1,014 millibars, weighs 1,253 gm., what would be the weight of a cubic
metre of air at 3,300 feet if the temperature at this elevation is 5° C.
and the pressure 900 millibars ?

CHAPTER VII

1. How is quantity of heat defined, and what is meant by a unit
of heat ?

What is meant by the statement that the calorifics value of a certain
gas is 450 B.Th.U. per cubic foot ?

2. Gas companies base their charges for gas entirely on the total
amount of heat which could be obtained from the gas supplied. A
certain company charged 14^ pence per therm (1 therm = 100,000
B.Th.U.). What would be the cost of heating the water for a bath
by means of a geyser, if the water entered the geyser at 60° F. and
came out at 110° F., the amount of water used being 24 gallons, i.e.
240 Ib. ? Assume that the whole of the heat from the gas went into
the water.

3. What is meant by the thermal capacity of a body ? Two hundred
gm. of water, at 30° C. , are poured into a metal vessel originally at 15° C. ,
the temperature of the room. The water is stirred and its temperature
is found to have dropped to 28-5° C. Calculate the amount of heat
communicated to the vessel and from this its thermal capacity.

4. Describe in detail any method by which the heat of combustion
of coal may be determined.

5. In a determination of the heating value of coal by means of a
Darling calorimeter tne following figures were obtained : —

Weight of coal burnt ....... 1-52 gm.

Initial temperature of the water in the calorimeter . . 13-20° C.

Highest temperature of water 20-65° C.

Mass of water in calorimeter ...... 1,400 gm.

Water equivalent of apparatus . . . . .210 gm.

Calculate the heating value of coal from these figures.

6. What is meant by the specific heat of a substance ? Is it neces-
sary to specify the unite of mass or the scale of temperature in giving
the value of the specific heat ?

Describe a method of determining the specific heat of a metal like
iron or tin.

7. A lump of lead weighing 752 gm. was heated to 100° C. and
then immersed in 500 gm. of water contained in a copper calorimeter.
The initial temperature of the water was 15-0° C. ; the final temperature
of the water was 18-6°C. The empty calorimeter weighed 200 gm.,

340 APPENDIX

and the specific heat of copper is -095. Calculate the specific heat
of lead.

8. Jn the experiment described in Question 7, the weighings were
made to the nearest gm., but the temperature of the water was
measured by means of a good thermometer reading to tenths of a
degree. Explain why it was necessary to measure the temperature
to such a degree of accuracy.

9. In order to determine the specific heat of aniline, a lump of copper,
weighing 320 gm., was heated to 100° C. and then immersed in 468
gm. of aniline contained in a copper calorimeter of which the weight
was 200 gm. The initial temperature of the aniline was 10° C. and
the final 19-5° C. Calculate the specific heat of aniline, taking that
of copper as -095.

10. In order to obtain an approximate value for the temperature
of a furnace, a piece of iron weighing 1 Ib. was left in the furnace for
some time and then quickly withdrawn and plunged into 6f Ib. of water
contained in a calorimeter of which the water equivalent was £ Ib.
The original temperature of the water was 13° C., and the final tempera-
ture (after adequate stirring) was 38° C. Calculate the temperature
of the furnace, taking the mean specific heat of iron over the expected
range as -15.

11. Describe how you would determine the specific heat of a liquid
by the method of cooling.

12. In a determination of the specific heat of a liquid a quantity
of the liquid was heated and placed in a metal vessel provided with
an adequate stirrer. The water equivalent of the vessel was 8 gm.,
and the mass of liquid 154 gm.

The rate of cooling was as follows : —

50° C.-450 C. required 167 seconds.

45°C.-40°C. „ 190 „

40°C.-35°C. „ 230 „

Repeating the experiment with an equal volume of water (mass = 150
£m.) in the vessel, the times were : —

50° C.-450 C. required 300 seconds.

45° C.-400 C. „ 350

40° C.-350 C. „ 405 „
Calculate the specific heat of the liquid.

CHAPTER VIII

1. What is meant by the term heat engine ?

In the working of heat engines there are certain broad principles
xjommon to all types. What are these principles ?

2. What is meant by (a) work, (6) power ?

What is meant by the brake-horse-power of an engine and how may
it be measured ?

3. In a test of a certain engine, a rope brake was used. A load of
80 Ib. weight was attached to one end of the rope and a spring balanc®

APPENDIX

341

to the other. The diameter of the wheel (measured to the centre of
the rope) was 4 feet, and with the engine running at 410 revolutions
per minute the reading of the spring balance was 16 Ib. weight. Calcu-
late the B.H.P. developed by the engine.

4. Describe the arrangement and working of a double-acting steam-
engine cylinder. Show how this action is an example of the general
processes characteristic of all heat engines.

5. Describe the action of an internal-combustion engine, working
on the four-stroke cycle.

6. What connection is there between heat and mechanical work ?
What is meant by the mechanical equivalent of heat ?

7. Describe a laboratory method of determining the mechanical
equivalent of heat and mention any precautions and corrections which
are necessary.

8. In an experiment to determine the mechanical equivalent of
heat, a brass drum containing 250 gm. of water was driven against
the friction of a silk belt, and the temperature was noted every minute.
At the end of eight minutes the drum was stopped, the ends of the
belt detached from their loads and fastened together, and the motor
started again ; the rate of cooling was noted. The following figures
were obtained : —

Room temperature . . 15° C.

Diameter of drum . . .6 inches.

Water equivalent of drum . . 39 gm.

Load on one end of belt . . 4,275 gm.

Load on other end of belt . . (473— spring balance reading)

WORKING

COOLING

Time
(Minutes).

Temperature.

Revolution
Counter.

Spring
Balance
(Ib.).

0

14-0° C.

5,184

•50

1

16-0

5,313

•50

2

17-9

5,441

•50

3

19-8

5,576

•50

4

21-6

5,713

•50

5

23-2

5,845

•45

6

24-8

5,979

•45

7

26-3

6,116

•45

8

27-8

6,249

—

Time
(Minutes).

Temperature.

8J

27-8° C.

9

27-4

10

26-8

11

26-2

12

25-6

13

25-1

14

24-6

15

24-1

16

23-7

Determine the total amount of heat supplied to the drum and ite
contents, making the necessary correction for cooling.

Determine also the work done against friction, and hence calculate
the mechanical equivalent of heat. (1 Ib. = 454 gm.)

9. The heat of combustion of a pint of petrol is about 9,600 T.U.
A certain engine was found to use 3 pints of petrol per hour when it
was developing 4 B.H.P. What fraction of the heat supplied was
converted to useful work ? (Mechanical equivalent of heat = 1,400
foot-pounds per T.U.)

342 APPENDIX

10. A certain car is found to run on the average 30 miles to each
gallon of petrol used. Assuming that the engine converts $ of the
heat supplied into useful work, calculate the work done in driving the
car 30 miles, and hence the average resistance which has to be over-
come when the car is moving.

11. In a very efficient steam engine, the consumption of coal per
B.H.P. was 1-25 Ib. per hour. The coal had a heating value of 15,000
B.Th.U. per Ib. Calculate the brake thermal efficiency of the engine.
(J = 778 foot-pounds per B.Th.U.)

12. The thermal efficiency of all practical heat engines is very low.
Why is this ? Is there any likelihood of improvement in design pro-
ducing efficiencies anywhere near 100 per cent. ? Give reasons for
your answer.

13. In determining the power of large engines the ordinary friction
brake cannot be used, and some other method (e.g. measurement of
shaft torsion) is adopted. Explain why this is so and illustrate your
answer by considering the heat which would be developed in the
case of a 10,000 H.P. engine.

14. The Mark VII bullet leaves the rifle with a speed of 2,440 feet
per second. The ordinary expressions of mechanics show that in the
absence of any air resistance a body projected vertically with this
velocity would rise to a height of about 93,000 feet. If the bullet
weighs f ounce, how much work does it do in the ascent ?

If the bullet were fired directly into a sandbank and the whole
energy converted to heat, what quantitv of heat would be gener-
ated?

15. What do you understand by the Principle of the Conservation
of Energy ?

Illustrate your answer by working the following example : Experi-
ment shows that when a rifle is fired vertically, using Mark VII ammu-
nition, the air friction is such that the bullet rises not quite 10,000
feet. When it falls again it reaches the ground with a velocity of
about 500 feet per second, i.e. the velocity it would acquire by falling
about 3,900 feet in the absence of air resistance.

Calculate the amount of heat which is generated (a) on the upward
journey, (6) on the downward ? How does the total amount compare
with that in Question 14 ?

16. Give an account of the experimental work which led to the idea
of the equivalence of heat and mechanical energy.

17. In scientific and engineering discussions, various units are
employed to measure work and energy, e.g. the foot-pound, the
erg (=1 centimetre-dyne), the joule (— 107 ergs), the kilogram-
metre. If the value of J is given as 778 foot-pounds per B.Th.U.,
calculate its value —

(a) in foot-pounds per T.U. ;
(6) in ergs per calorie ;

(c) in kilogram-metres per large (kilogram) calorie.
(2'54 cm. = 1 inch, g may be taken as 981 cm. per sec. per sec., so
that a force of 1 gm. weight represents 981 dynes.)

18. Mention the chief general methods which have been used in
the determination of J, and describe one method in greater detail.

APPENDIX 343

19. Describe a "continuous-flow " method of determining J, such
as that of Reynolds and Moorby, or Callendar and Barnes. Mention
any advantages of using a continuous flow.

20. Electrical power is usually measured in watts (1 watt =10*
ergs per second). An electric kettle holds 1% pints of water and is
stated to take 350 watts. How long should it take to raise the tem-
perature of the water from 15° C. to 100° C. if 80 per cent, of the heat
is utilized ? (Take the water equiv. of the kettle as £ Ib. ; 1 pint
of water weighs 1 J Ib. ; J = 4-2 X 107 ergs per calorie, 454 gm. = 1 Ib.)

21. In accurate scientific work it is necessary in defining a unit
of heat to specify the particular range of temperature, e.g. the 15-
degree calorie.

Mention any experimental work which shows that this precaution
is necessary.

CHAPTER IX

1. Comment on the following, and describe any experiment to
illustrate the cause of the phenomena :

(a) Ice floats on water.

(6) After a very cold night the water-pipes of a house are some-
times found to have burst.

(c) The action of frost in breaking up soil and rocks.

2. A lump of cast iron will float in the molten liquid, but in the
case of most metals, e.g. copper, the solid sinks in the liquid. Explain
clearly why it is possible to obtain sharp castings with iron but not
with a metal like copper.

3. A piece of ice at — 5° C. is gradually heated. Illustrate, with
the aid of a graph, how the volume changes as the temperature rises
from -5°C. to 15° C.

4. Large masses of snow often remain quite a long time after a thaw
has set in, especially if they are not exposed to wind ; e.g. snow is
usually to be found in the gullies of the north face of Great End (Lake
District), even in May. Why is fehis ?

5. What is meant by the latent heat of fusion of a substance ?
What quantity of heat would be required just to melt 2 kilograms

of lead originally at 15° C. ?

(Specific heat of lead = -031, melting-point 327° C. Latent heat
of fusion, 5 calories per gramme.)

6. A calorimeter (water equivalent 20 gm.) contains 500 gm. of
water at 20° C.

What weight of dry ice originally at 0° C. must be added to the
water in order that the temperatures may be reduced to 10° C. ?

What would be the result of adding 160 gm. of ice ? (Latent heat
of fusion of ice = 80 calories per gm.)

7. What is meant by the melting-point of a solid ? Illustrate your
answer by describing what happens when a solid is gradually heated
from a temperature well below the melting-point to one well above
it, taking as examples ice and paraffin wax.

344 APPENDIX

8. Describe in detail a method of determining the melting point
of a solid.

9. Skating becomes very difficult if the temperature is greatly below
freezing-point — the skate will not " bite." Explain why this is so,
and describe an experiment to illustrate your answer.

10. It is usual to define 0° C. as the temperature at which pure ice
melts rather than the temperature at which pure water freezes. Can
you suggest any reason for this ?

11. In what way is the melting-point of a solid affected by pressure ?
Illustrate your answer by reference to the following points :

(a) When one is walking on snow a hard mass more or less of ice
is apt to be formed under the heel of one's boot, but it does not do
so if the temperature is very low.

(6) The temperature at points in the earth far below the surface
must be above the melting-point of any known substance, but there
is every reason to believe that the interior of the earth is not liquid.

(c) In determining the lower fixed point of a thermometer, great
care is taken to have the ice pure, but no correction for the barometer
reading is required (as far as the temperature is concerned)

12. Describe the behaviour of a mixture of ice and salt. In the
coiirse of your answer deal with the following point :

To make a very fair freezing mixture salt can be added to ice ; a
good way of melting ice on a frozen doorstep is to add salt to it.

13. Small pieces of ice at — 10° C. were added (with constant stirring)
to 200 gm. of water contained in a calorimeter of water equiv. 10 gm.
The original temperature of the water was 18° C., and when it had
dropped to 8° C. the whole was weighed and it was found that 22-6
gm. of ice had been added. Calculate the specific heat of ice, taking
the latent heat of fusion as 80 calories per gm. Would you regard
this as an accurate method of finding the specific heat ? Give reasons
for your answer.

CHAPTER X

1. What do you understand by the boiling-point of a liquid ? Illus-
trate your answer by describing what happens when a flask containing
some water, and fitted with a thermometer and outlet tube, is heated
over a burner.

2. Describe experiments which illustrate how the boiling-point of
a liquid depends on the pressure to which it is exposed.

3. What is meant by the latent heat of vaporization of a liquid ?
Describe a method of determining the latent heat of vaporization

of water at 100° C.

4. Twenty gm. of steam at 100° C. are passed into 500 gm. of water
at 15° C. contained in a calorimeter of water equivalent 20 gm. Cal-
culate the final temperature of the water. (Latent heat of vaporiza-
tion of water at 100° C. = 539 calories per gm.)

5. A certain boiler worked at a pressure of 160 Ib. per sq. inch.

APPENDIX 345

At this pressure water boils at 188° C. If the feed-water entered the
boiler at a temperature of 90° C., it was found that 11 Ib. of steam were
produced for each pound of coal burnt. Taking the latent heat of
vaporization of water at 188° C. as 477 T.U. per Ib. and the heating
value of the coal as 8,400 T.U. per Ib., calculate what percentage
of the heat supplied has gone into the steam.

6. Exhaust steam from an engine passes into a surface condenser,
i.e. a vessel in which are a number of tubes through which cooling
water circulates : the steam is condensed by contact with the outside
of these pipes, but does not mix with the cooling water. The prescure
in the condenser is 4 Ib. per square inch absolute, at which pressure th«
temperature of the steam is 67° C. and the latent heat of vaporization
is 558 T.U. per Ib. A steady state is reached in which the cooling water
enters the pipes at 15° C., and leaves at 30° C., while the water from
the condensed steam (which is pumped out) is at 50° C. Calculate
the amount of heat which has to be abstracted from each pound of
steam to condense it to water at 50° C., and hence the number of pounds
of cooling water required per pound of steam.

7. Observations of the boiling-point of water on the summit of a
mountain enable the height of the mountain to be deduced. Explain
why this is so and describe a suitable apparatus for making the obser-
vation.

8. What is meant by evaporation ?

Illustrate your answer by considering what happens when an open
vessel containing some liquid is placed in a large enclosure.

9. What is meant by the saturated vapour pressure of a liquid ?
Describe a method of determining the vapour pressure of water at
different temperatures between say 15° C. and 100° C.

10. What is meant by ebullition or boiling ?

What is the characteristic feature which distinguishes boiling from
ordinary evaporation ?

Why cannot a liquid boil until its temperature is such that the
vapour pressure is equal to the pressure to which the liquid is exposed ?

11. How would you determine the boiling-point of a liquid of which
only a small quantity is available T

12. How can the vapour pressure of liquids at high temperatures
be determined ?

13. A dish of water is placed in an enclosure. Compare (a) the rate
of evaporation, (6) the final pressure of water vapour in the enclosure
in the three following cases: (1) when the enclosure at the beginning
is vacuous, (2) the enclosure originally contains dry air, (3) the enclo-
sure contains air with a certain amount of water vapour already in
it.

14. Why do puddles of water on a road dry up more quickly when
a wind is blowing than they do in still air T

15. One is almost certain to catch a chill if one sits about in wet
clothes, especially f there is a draught ; on the other hand, it is quite
safe (but rather uncomfortable) to get straight out of a cold bath and
dress without drying. Explain why this is so.

16. Large masses of snow are often seen to diminish perceptibly

346

APPENDIX

111 size after being exposed for some time to the action of wind, even
though the temperature all the time is below freezing-point.
Explain why this is so.

17. What is meant by an unsaturated vapour ? Is there any real
distinction between the physical properties of unsaturated water
vapour and a gas like sulphur dioxide at ordinary temperatures ?

18. At one time, gases like oxygen and hydrogen were referred to
as " permanent gases " to distinguish them from gases like sulphur
dioxide and chlorine, which could be liquefied by pressure. Is there
any real difference between the so-called permanent gases and the
others, and if there is any difference, in what does it consist ?

19. If a stream of air is blown through ether in an open flask, a very
marked cooling effect is noted. Explain clearly why this is so. A
similar process with water instead of ether produces a much smaller
cooling effect, although the latent heat of vaporization of water is
greater than that of ether. Why is this ?

20. Describe and explain the general principles on which an ammonia
refrigerating machine works.

21. Describe the construction of Bunsen's ice calorimeter and explain
how it can be used to measure accurately the specific heat of a metal.
What are the special advantages of a calorimeter of this type ?

22. Describe a form of Joly's steam calorimeter, and mention any
advantages of the apparatus.

A piece of aluminium, weighing 12-32 gm., was placed in the
chamber of a steam calorimeter. The initial temperature of the
aluminium was 15° C. Steam was then passed in, and when a steady
state was reached it was found that the weight of steam condensed
(after allowing for buoyancy) was -408 gm. Taking the latent heat
of vaporization of water as 539 calories per gm., calculate the specific
heat of aluminium.

23. Describe a method of determining the specific heat of a solid
at very low temperatures.

CHAPTER XI

TABLE OF SATURATED VAPOUR. PRESSURE or WATER

Temperature. *CC.

2°.

4°.

6°.

8 •

10°.

12°.

14°.

16°.

18°.

20'.

Vapour pressure,

mm. of mer-

cury .

5-3

6-1

7-0

8-0

9-2

10-5

12-0

13-6

15-5

17-5

1. If a vessel containing a cool liquid (e.g. a tankard of beer from
a cool cellar) be brought into a warm room, the outside of the vessel
becomes moist although no liquid has been spilled. Explain how thjf
happens,

APPENDIX 347

2. What is meant by the relative humidity of the air ? On a hot
day in Bummer the air feels dry, but the amount of water vapour in
a given volume of it (as determined by passing it through a weighed
' ' drying ' ' tube) may be greater than that in the same volume of air
on a damp, cold day. Why is this ?

3. Explain how a deposit of dew is formed. Why is dew much
more likely to form when the air is still than when a wind is blowing ?
Why does it form more readily on objects near the ground than on
those which are at a higher elevation ?

4. A clear still night, following a hot, dry day, is a condition very
favourable to the formation of dew. Why is this such a favourable
condition ?

5. Describe either Dine's or Regnault's hygrometer ; explain how
you would use it to determine the dew-point, and point out any
precautions which are necessary.

6. Explain clearly how to calculate the relative humidity of the
air when the dew-point has been determined. » .

7. On a certain day the temperature of the air is 16° C. and the
dew-point is 8° C.

Calculate the relative humidity of the air.

8. On a certain day, the temperature is 18° C. and the relative
humidity is 50 per cent. ; on another day the temperature is 6° C.
and the relative humidity 90 per cent. Which day would feel to be
the drier, and on which day would the amount of water per unit volume
be the greater ?

9. Describe the wet-and-dry bulb hygrometer and explain how it
works. Why is this instrument used in meteorological stations rather
than one of the dew-point type such as Dine's ?

10. Describe how the wet-and-dry bulb thermometer is modified
in modern work so as to make it less uncertain.

11. What is the principle of Crova's dew-point hygrometer ? Why
is this a great improvement on earlier dew-point instruments ?

12. A quantity of gas is collected over water, the temperature being
14° C. and the pressure to which it is exposed being 758 mm. of mercury.
What would be the pressure of the gas if all the water vapour were
removed from it and it occupied the same volume as it did when
wet ?

13. On dissolving a piece of metal in acid, hydrogen was given off ;
it was collected over water and the volume found to be 500 c.c. The
temperature was 14° C. and the pressure 765 mm. of mercury.

If one litre of dry hydrogen at 0° C. and 760 mm. weighs -09 gm.,
what was the weight of hydrogen given off by the action of the metal ?

14. Air is contained in a closed vessel which has also in it a quantity
of water.

If the pressure in the vessel is 760 mm. when the temperature is
4° C., what would be the pressure when the temperature is 20° C. ?

15. A mixture of air and water vapour (unsaturated) is cooled (a) at
constant volume, (6) at constant pressure. -How does the pressure
of the aqueous vapour change hi each case ; and in which will the
temperature at which condensation first occurs be the higher ?

348 APPENDIX

16. Describe a method of determining the humidity of the air by
direct weighing with a chemical hygrometer.

17. In a chemical hygrometer, air was drawn through drying-tubes
into an aspirator. The volume of air which came into the aspirator
was 5 litres, the temperature in the aspirator being 12° C. The increase
in weight of the drying-tubes was -045 gm. The temperature of the
air outside the apparatus was 16° C. and the barometer reading 750
mm. Calculate the pressure of aqueous vapour in the air, and the
relative humidity. (Density of aqueous vapour = -622 that of air
at the same temperature and pressure ; weight of a litre of air at
0° C. and 760 mm. = 1-293 gm.)

18. Why are fogs more prevalent in smoky cities than in the open
country ?

19. A drop of water is placed in an enclosure initially free from water
vapour. How is the rate of evaporation affected by (a) the curvature
of the surface of the drops, (6) the purity of the water, (c) the pressure
of the air in the enclosure ? How do these three factors affect the
final pressure of the vapour if the enclosure is so small that the drop
does not completely evaporate T

CHAPTER XII

1. In what different ways can hoat be transmitted from one place
to another ?

Give an example of each process and point out what are its char-
acteristic features.

2. When heat is transmitted by conduction through a sheet ot
material, upon what factors does the rate of transmission depend ?

3. The thermal conductivity of aluminium in C.G.S. units is -5.
What quantity of heat would be transmitted per minute through a
sheet of aluminium 1 square metre in area and 3 cm. thick if the opposite
faces are at temperatures differing by 30° C. ?

4. If the thermal conductivity of aluminium is -5 in C.G.S. units,
what is the conductivity when the unit of heat is the B.Th.U., the unit
of length is a foot, and the Fahrenheit scale of temperature is used ?
(1 inch = 2-54 cm. ; 1 Ib. = 454 gm.)

5. Describe a method of determining the thermal conductivity of
a metal like copper.

6. Give an account of the construction and working of a Davy safety
lamp, and describe any experiment to illustrate its action.

7. How would you show that water is a very poor conductor of
heat ?

8. Why is material like cotton wool such a very poor conductor of
heat ?

9. Why does a hot object cool more rapidly when exposed to a
draught than it does in still air ?

10. When water is heated in a metal vessel over a flame, the tern-

APPENDIX 349

perature of the outer surface of the vessel is very much lower than
that of the flame. Why is this ?

11. A piece of paper is wrapped tightly round a bar, one-half
of which is of brass and the other half of wood. The bar is then held
over a flame and it is found that the paper is charred where it is in
contact with the wood, but not where it is in contact with the brass.
Explain clearly how it is that this happens.

12. In the design of boiler furnaces, the problem is not so much
to get the heat to flow through the metal of the furnace tube as to get
it to flow from the furnace gases to the metal. Illustrate this point
by determining what must be the difference in temperature between
the two sides of a steel furnace tube 13 mm. thick if the quantity of
water evaporated was 200 kilograms per hour per square metre of
surface (a very high rate). Take the latent heat of vaporization as
539 calories per gm. ; conductivity of steel '11 C.G.S. un ts.

13. Explain why it is dangerous to allow large deposits of scale to
accumulate in boilers.

14. What connection is there between the thermal and electrical
conductivities of metals ?

15. How could you compare the thermal conductivities of two
metals if they were provided in the form of bars of the same size and
shape ?

16. Describe Lee's method of determining the conductivity of poor
conductors.

17. Describe a method of determining the conductivity of a liquid.

18. The surface of a pond is in contact with air, at temperature
below 0° C.,and ice is forming at the surface. Assuming that the tem-
perature of the air remains constant, and the water is all at 0° C., find
how the rate of increase in thickness of the ice formed varies with
the time.

CHAPTER XIH

1. What do you understand by the statement that heat is trans-
mitted from one place to another by convection ?

Illustrate your answer by describing the action of a system of heating
by hot water pipes.

2. Describe the ordinary thermo-syphon system of cooling the
engine of a car and explain how it is to some extent self -regulating.

3. Explain the action of a chimney as a means of producing a draught.

4. Explain why a room heated by an open fire is more pleasant
and less " stuffy " than one heated by radiators.

5. Describe what you would consider to be a satisfactory method
of ventilating a large hall.

6. A chimney is 200 feet high, and the mean temperature of the
gases in it is 250° C. Determine the difference in pressure between
the inside and the outside of the chimney at a point near tho base.

350 APPENDIX

if the temperature and pressure outside are 15° C. and 760 mm. It
may be taken that 24 Ib. of air pass through the fire for each pound of
fuel burnt, and the volume of the products of combustion is about the

same as that of the air used, i.e. the density of the flue gases is _

that of air at the same temperature and pressure; and the weight of
a cubic foot of air at 0°C. and 760 mm. as '08 Ib.

CHAPTER XIV

1. What is meant by the statement that a body is losing heat
by radiation ?

Describe any experiments to illustrate the properties of radiation,
e.g. how it travels, the way it is reflected and refracted.

2. If two similar thermometers are taken, and the bulb of one is
coated with lampblack, it is found that when the two are exposed
to sunlight the black-bulb thermometer registers the higher tempera-
ture. On the other hand, if the two thermometers are heated and
then allowed to cool, the black one cools more rapidly. Explain why
this is so, and describe experiments to illustrate your explanation.

3. How is it that on snow-slopes at high elevations one is often
very warm in the sunshine while the temperature of the air is much
below freezing point and very little melting of the snow occurs ?

4. In gardens, fruit trees, e.g. peaches, are often planted beside a
wall which faces south. How does the wall assist their development ?

5. What connection is there between the emissive and absorbing
powers of a surface ?

Describe any experiments to illustrate the connection.

6. Describe the construction of a thermopile on some other instru-
ment suitable for detecting and measuring radiation.

7. What reasons have we for supposing that light and the invisible
radiation from a hot object are essentially similar ?

8. A hot object, such as a metal vessel or a copper ball at 200° C.
or 300° C. (i.e. not nearly hot enough to be luminous), is placed in
front of a thermopile. What is the effect of interposing (a) a sheet of
glass, (&) a plate of rock salt, (c) a solution of iodine in carbon disulphide
contained in a trough of very thin glass ? If an ordinary optical
lantern using an arc lamp, and with all its lenses (including the thick
condenser) in, be directed on to a thermopile, a considerable effect is
produced. How can this fact be reconciled with the results of the
experiments of the first part of the Question ?

9. The radiation from an arc lamp differs in quality from that given
off from a solid at a temperature of about 250° C. Illustrate what is
meant by this statement by considering the behaviour of a narrow
beam of the radiation from each of these sources when passed through
a prism of rock salt.

10. In the illustration of Question 9, if a prism of glass were used
instead of rock salt, what difference would there be in the observe^
effects ?

APPENDIX 351

11. Describe the construction of a thermos flask and explain why
this construction is particularly effective in obtaining heat insulation,

12. Write short notes on the following practical effects :

(a) A glass fire screen will cut off a large part of the heating effect
of a fire while allowing the fire to be seen.

(6) On a sunny day the temperature inside a glass -roofed building,
e.g. a greenhouse, is higher than it would be with an opaque roof.

(c) A clear night is usually colder than a cloudy one.

13. In an electric incandescent lamp a current is sent through a
thin filament which in consequence becomes hot and gives out light*
Upon what conditions wiH the proportion of the electrical energy
which is converted to light depend ? What becomes of the remainder
of the energy ? Why can a gas-filled tungsten-filament lamp be made
more efficient tfofun a carbon lamp ?

CHAPTER XV

1. Give an account of Prevost's Theory of Exchanges, and explain,
in terms of this theory, what happens when a piece of ice is brought
near a thermopile in an ordinary room.

2. What do you understand by the " full " radiation corresponding
to a particular temperature ? Why is the radiation which escapes
from a small aperture in a uniform-temperature enclosure taken as
equivalent to the radiation from a perfectly black body at that tem-
perature ?

3. Experiment, shows that the emissive power of a surface is pro-
portional to its absorbing power. Show how this result follows from
the theory of exchanges.

4. Radiation of almost any quality passes readily through rock
salt, but the radiation from another piece of heated rock salt is strongly
absorbed. Show that this is a case of a general result which is to be
expected.

5. State Newton's Law of Cooling. To what conditions was Newton
referring when he enunciated the law ? Under what conditions can
it be used to give the rate of cooling of a body sufficiently accurately
for practical purposes ?

6. State Stefan's Law for the total radiation from a body. In what
conditions can it be taken to represent what will actually happen ?

7. A body with a dull black surface is placed in an enclosure main-
tained constantly at 0° C. If conduction and convection are prevented
(e.g. by having a very high vacuum in the enclosure), compare the
rate ofVooling of the body when it ia »t 200° C. with its rate when it
ia »t 100° a

352 APPENDIX

CHAPTER XVI

1. Describe the construction of (a) steel-bulb mercury thermometers,
(6) vapour-pressure thermometers, and point out the conditions for
which they are suitable.

2. Give an account of the ways in which temperature can be measured
electrically.

3. In what way can very low temperatures, e.g. — 150° C. to — 250°
C., be measured ?

4. Give an account of the methods available for measuring very
high temperatures, e.g. 2,000° C.

5. In what way is the temperature of a body as recorded by a radia-
tion pyrometer affected by the nature of the surface of the body ?

6. How are pyrometers calibrated (a) for temperatures up to 1,600°
C., (6) above that temperature ? If a pyrometer registers the tem-
perature of a body as 2,000° C., what precisely does this mean ?

7. Give an account of the methods by which the temperature of the
sun has been estimated.

PART II
CHAPTER XVII

1. Mention some of the chief experimental work which has been
done on the relation between the pressure and the volume of a gas
and give an account of the nature of the deviations from Boyle's Law
observed in the case of gases such as oxygen, nitrogen, and hydrogen,
and also of carbon dioxide.

2. Give an account of Andrews' work on the isothermals of carbon
dioxide, and show on a rough diagram the shape of the isothermals. By
the aid of your diagram explain what is meant by the critical tempera-
ture of a substance.

3. If you had available a diagram showing a series of isothermals
of a substance, e.g. ether, how would you plot a graph showing the
saturated vapour at different temperatures ?

4. Discuss the meaning of critical temperature and show how it is
possible for a substance to pass without any abrupt change from a
state which is undoubtedly gaseous to one which is undoubtedly liquid.

CHAPTER XVIII

1. Explain clearly how the pressure exerted by a gas is accounted
for on the kinetic theory, and show that the theory is consistent with
Boyle's Law and Avogadro's Hypothesis.

2. Give an account of Van der Waal's modification of the kinetic

APPENDIX 353

theory so as to include the mutual action of the molecules and their
finite dimensions. Show that this modification is consistent with the
observed deviations from Boyle's Law. Show how the constants in
Van der Waal's equation are connected with the critical constants of
a substance and hence how an indication of the critical temperature
of a gas can be deduced from observations of its deviation from Boyle's
Law.

CHAPTER XLX

1. Describe a method of determining the specific heat of a gas (a) at
constant volume, (b) at constant pressure. Why is the latter specific
heat necessarily greater than the former ?

2. In some stationary steam-engine plants, the feed -water for the
boiler is heated by causing it to flow through tubes over which the
gases from the furnaces pass on their way to the chimney, the arrange-
ment constituting an " economizer." Assuming that the flue gases
reach the economizer at a temperature of 350° C. and leave it at 200°
C., calculate the amount of heat saved per pound of coal. Take the
specific heat at constant pressure of the flue gases as -21 ; that 75
per cent, of the heat from the gases goes into the water ; that for
each pound of coal 24 Ib. of air are used, so that 25 Ib. of flue gases
are formed. Express the result as a percentage of the calorific value
of the coal, which is 8,000 T.U. per pound.

3. Give an account of experiments to determine whether any internal
work is done when a gas expands, and describe the results obtained.

4. In the Joule and Thomson porous-plug experiment a heating
effect was obtained with hydrogen. Explain clearly what this indicates.

5. Describe in outline the modern method of liquefying air by the
regenerative cooling method.

6. Describe the method of liquefying hydrogen and point out why
the arrangement used in the case of air must be modified for hydrogen
and helium and in what the modification consists.

CHAPTER XX

1. What is meant by an adiabatic change ? Show that if a certain
quantity of gas increases its volume by a given amount, the fall in
pressure will be greater for an adiabatic change than for an isothermal
one.

2. What is meant by a " perfect " gas ?

If pv = RT is the general equation for a perfect gas, show that
(a) J(Cp — Cv) = R where Cp and Cv are the thermal capacities of the
gas at constant pressure and volume respectively. (6) The equation

for an adiabatic change is pvy = constant where y = Cp/Cv.

3. In a Diesel engine, air alone is taken into the cylinder on the
induction stroke and is compressed on the next stroke. The tempera-
ture is raised so much by the compression that when the fuel is sprayed

A A

354 APPENDIX

in it automatically ignites. Assuming that air is taken into a Diesel
cylinder at 27° C. and 15 Ib. per square inch and is compressed adiabati-
cally until at the top of the stroke the pressure is 500 Ib. per square
inch, calculate (a} the compression ratio, i.e. the ratio of the volume
available for the air, when the piston is at the end of its outstroke, to
the volume when the piston is at the end of its instroke; (6) the
temperature of the air at maximum compression.

4. The volume elasticity of a substance is denned as the ratio of
the increase in pressure dp to the corresponding strain for com-
pression. Show that the elasticity of a gas for an adiabatic compression
is equal to yp. Take 7 as 1'40.

Hence show how the value of 7 may be deduced from a determina-
tion of the velocity of sound in a gas.

5. Give an account of the methods of determining 7.

How does a knowledge of 7 give an idea of the number of atoms iu
the molecule of the gas ?

CHAPTER XXI

1. What is meant by an indicator diagram for an engine and what
information does it give ?

2. What is the First Law of Thermodynamics ? What is the
essential condition in order that heat may be continuously converted
to work by a heat engine ?

3. What is meant by a reversible cycle ?

Describe Carnot's cycle ; upon what does the efficiency of this cycle
depend ?

4. Show that for engines working between the same temperatures
no engine can be more efficient than one working on a reversible cycle.
How does this proposition lead to the idea of a scale of temperature
independent of the properties of any substance ?

5. Explain how the absolute thermodynamic scale of temperature
is defined, and show that it coincides with the perfect gas scale.

CHAPTER XXII

1. Give an account of the Principle of the Conservation of Energy
and the Principle of the Dissipation and Degradation of Energy.

2. What is meant by the entropy of a substance ? Show that the
entropy of a substance in a certain state depends only on that state
and not on the way it was reached. Calculate the entropy of a pound
of saturated steam at 100° C. and 760 mm., taking the entropy of water
at 0° C. and 760 mm. as zero.

3. If the working substance in an engine is taking in heat from a
source at a temperature 0, and if we know that the entropy of the
substance in the initial and final states is respectively 0t and 02, what
does this tell us about the amount of heat absorbed by the substance t
Explain carefully what your statement means.

APPENDIX 355

ANSWERS

CHAPTER II

6. 36-9° C. ; 176-7° C. ; - 12-2° C. ; - 23-3° C. ; 25* C. 7. 69°
F. ; 5° F. ; — 459-4° F. ; 54-5° F.

CHAPTER III

1. About 3-89 inches. 2. 1-00038 metres. 3. 40-51 cm. 4. 0-136
inches. 5. Expansion of the steel calipers themselves for a range of
20° C. would make a difference of -0022 cm., i.e. about 315 mm. on a
10 cm. length. Thus maximum uncertainty would be of the order
of the degree of accuracy of the reading and it would hardly be worth
correcting for the expansion of the scale. 8. 5-448 tons weight. 9.
Not less than 464° C. 10. (a) 3-25079 cm. (6) 3-24943 cm.

CHAPTER V

2. -000159. 3. -00107 (range 0°C. to 50°). 5. -0000235. 6. -0000015.
7. (a) -82 per cent., (6) -28 per cent. 10. -00110 (range 0°-30° C.). 13.
•000181. 14. An error of -01 cm. represents -84 per cent, of 1-18 cm.
and therefore -84 per cent, of the result. Expansion of silica repre-
sents -82 per cent, of expansion of mercury (Question 7). Hence
reading of difference of heights must be correct to at least -01 cm.
15. 75-313 cm.

CHAPTER VI

1. £ cubic foot or 288 cubic inches. 2. 6f cubic feet less T*T cubic
foot which would still be in when the pressure became atmospheric.
4. 75-45 cm. 7. 77-6 Ib. per square inch. 8. -00366. 11. 58-1° C. 13.
(a) ~- 273° C., (6) - 459-4° F. 14. 419-6 c.c. 15. -005309 Ib. 16.
1,128 gm.

CHAPTER VII

2. 1-73 pence nearly. 3. 300 calories; 22-2 calories. 5. 7,893
calories per gm. 7. -0305. 9. -51. 10. 1205° C. (i.e. about 1200°
C.). 12. -.52.

CHAPTER VIII

3. 10 B.H.P. (9-991). 8. 10-63 T.U. ; 14,810 foot Ib. ; J = 1,393
foot Ib. per T.U. [Note. — There is an uncertainty in the cooling cor-
rection depending on the graph which might be as much as -15° C.,
making an uncertainty of about 1 per cent, in the result.] 9. About
•2 (-1964). 10. About 136 Ib. weight. 11. 13-57 per cent. 13. Heat
developed at the rate of over 235,000 T.U. per minute. 14. 2,325
foot Ib. ; 1-661 T.U. 15. (a) 1-482 T.U., (6) -109 T.U. The total
differs from 1-661 by -07 T.U., the equivalent of the energy of a bullet
moving at 500 feet per sec. 17. (a) 1400-4, (6) 4-187 x 107, (c)
426-8. 20. Twenty and a half minutes,

356 APPENDIX

CHAPTER IX

5. 29,344 calories. 6. 57-7 gm. ; the temperature would be reduced
to 0° C. and about 30 gm. of ice left unmelted. 13. -49.

CHAPTER X

4. 38-1° C. 5. 75-28 per cent. 6. 575 T.U. ; 38-3 Ib. of cooling
water. 22. -21.

CHAPTER XI

7. «59 nearly. 8. The warm day would feel to be the drier, but
the amount of water vapour per unit volume would be greater than
on the cold day. 12. 746 mm. of mercury. 13. -04241 gm. 14.
815 nun. 15. Temperature higher in case (6). 17. 8-9 mm. (8*896) ;
Rel. Humidity -654.

CHAPTER XII

3. Three million calories. 4. -0336 B.Th.U. per square foot per
1° F. per foot temperature gradient. 12. 35-4° C. 18. Rate of for-
mation of ice « rate of flow of heat through ice sheet and since
temperature difference between faces of ice sheet is constant, the

temperature gradient oc 1. Thus ~ = K'l .'. — <* 1 where

x dT x dT Vf~+~c

T is the time, and c a constant depending on the moment from which
T is measured.

CHAPTER XIII

6. 6-47 Ib. per square foot.

CHAPTER XV

7. Rate of cooling at 200 C. = 3-225 times that at 100° C.

PART II

CHAPTER XIX

2. 7-4 per cent.

CHAPTER XX

3. (a) 12-25, (b\ 544° C.

INDEX

Absolute expansion, 53, 62
Absolute temperature on the gas

scale, 80 ; on the thermody-

namic scale, 316
Absolute zero, 80, 269, 317
Absorbing power of surfaces, 214
Adiabatic changes, 296
Adiabatic equation, 296
Amagat, 258
Andrews, 259
Apparent expansion, 53
Avogadro's hypothesis, 270

Boiling, 159

Boiling point, determination of,

160 ; effect of pressure on, 152 ;

of a solution, 160, 161
Bomb calorimeter, 90
Boltzmann, 240
Boyle's Law, 70 ; deviations

from, 164, 257
Brake horse-power, 1 10
Bumping, 160

Cagniard de la Tour, 264
Cailletet, 289

Calibration of pyrometers, 253
Calendar's apparatus for J, 111
Callendar and Nichols, 198 ; and

Barnes, 127

Callendar's steam tables, 329
Caloric theory, 120
Calorific value of fuels, 89
Calory, 86, 128

Calorimeters, 89, 90, 92, 167, 168
Camot's cycle, 310
Chemical hygrometer, 1~9
Chimneys, 205
Clausius, 314
Clinical thermometer, 19
Clouds, rate of fall of, 183
Coefficient of linear expansion, 28 ;

measurement of, 29, 32
Coefficient of cubical expansion,

51 ; of liquids, 53, 62, 69 ;

gases, 73, 81
Cohesion, 266

Compensated leads for pyro-
meters, 246

Compensating devices, 45, 49
Condensation nuclei, 183
Conductivity of solids, 188 ; of
liquids, 194, 200 ; of gases, 194 ;
of bad conductors, 198
Constant pressure gas thermo-
meter, 79

Continuous spectrum, 223
Cooling correction, 96, 114
Cooling due to evaporation, 164
Critical temperature, 164, 259, 263
Cross tubes in boilers, 201
Crova's hygrometer, 178
Cyclical process, 308

Dalton's Laws, 156, 174
Dani ell's hygrometer, 175
Darling calorimeter, 92
Davy safety lamp, 197
Degradation and dissipation of

energy, 324
Delayed boiling, 160
Dew, 172
Dew point, 175
Dewar, 291
Diathermanency, 218
Dilatometer, 53
Dines' hygrometer, 175
Disappearing filament pyrometer.

249

Domestic hot water supply, 204
Dulong and Petit, 62, 238

Ebullition, 159

Efficiency of electric lamps, 229 ,

of a heat engine, 119, 312
Emissive power, 214
Entropy, 324 ; of steam, 330
Entropy-temperature diagrams,

331

Evaporation, 155
Expansion of glass vessels, G8

F6ry radiation pyrometer, 247
First Law of Thermodynamics, 308

357

358

INDEX

Fixed points, 13, 14

Fog, 183

Forbes, 198

Foster pyrometer, 248

Fourth power law, 238, 240

Freezing mixture, 144

Fused silica, 59

y, 280, 298, 301

y relation to number of atoms in
molecule, 305

Gas constant, 294 ; Gas engines,
107

Gas-filled lamps, 229

Gas laws, 270, 294

Gas thermometers, 79, 254

Glass, expansion of, 42, 68 ; selec-
tive absorption of, 231

Graham's Law of Diffusion, 270

Griffiths, E. H., 104, 127

Hampson, 291

Heat, units of, 86, 128

Heat engines, 104, 308

Heat pump, 312

Heights of mountains, 154

Him, 123

Hoar frost, 173

Hot air engine, 105

Hot water heating, 202

Hot water supply, 204

Hypsometer, 15, 154

Hydrogen thermometer, 79, 254

Indicator diagram, 306
Indicated horse power, 301
Ingen Hausz, 191
Infra red radiation, 226
Internal combustion engine, 107
Internal work when a gas expands,

285

Invar, 49
Inversion of the Joule Thomson

effect, 289

J, 111

J, determination of, 111, 124, 126
Joly's steam calorimeter, 283

Joule, 111, 122, 126; and Play
fair, 62 ; and Thomson, 285

Kammerlingh Onnes, 291
Kelvin, 314 ; and Joule, 285
Kelvin's Absolute Scale, 316

Latent heat of fusion, 131, 135 ;

of evaporation, 149
Lavoisier and Laplace, 38
Lees, 198
Leslie cube, 214
Linde, 291

Liquefaction of gases, 265, 289
Lummer and Pringsheim, 240

Mason's hygrometer, 177

Maximum density of water, 59, 62

Maximum and minimum ther-
mometer, 20, 21

Maxwell's Law, 271

Mayer's calculation of J, 284

Mechanical equivalent, 111

Melting point, 136, 138 ; effect of
pressure on, 142

Micelescu, 126

Mist, 183

Modern hygrometers, 178

Newton's Law of Cooling, 116, 237

Nickel steel, 49

Nitrogen thermometer, 254

Oil engine, 107
Optical pyrometer, 248
Olszewski, 289

Partial pressures, 174

Pictet, 289

Planck, 241

Porous plug experiment, 285

Prevost's theory of exchanges,

233

Pyknometer, 58
Pyrometer, electrical, 244, 245 ;

total radiation, 247 ; optical,

248 ; steel bulb, 243 ; vapour

pressure, 243
Pyro heliometer. 250

INDEX

359

Quality of radiation, 224, 226

Ralio-activity, 7.

Radiation from bodies at different

temperatures, 226
Radiators, 202, 203
Rate of cooling, 237
Ratio of specific heats, 280, 298,

301
Ratio of thermal and electrical

conductivities, 194
Refrigerating machines, 166
Regelation, 143
Regenerative cooling, 289
Regnault, 66, 95, 151, 162, 280
Regnault's hygrometer, 176
Relative humidity, 172
Resistance pyrometer, 244
Reynolds and Moorby, 126
Reversible process, 309
Rowland, 126

Scale in boilers, 196

Scale of temperature, 13, 22, 253,
316

Schuster and Gannon, 127

Screening action of clouds, 231

Screen, glass fire-, 231

Searle's apparatus for thermal
conductivity, 189

Second Law of Thermodynamics,
314

Silica vessels, 59

Specific heat, 87 ; of solids, 92 ;
liquid, 98 ; of gas at constant
pressure, 280 ; of gas at con-
stant volume, 282

Specific heats, ratio of, 280, 298,
301

Sound, velocity of, 298

Steel bulb mercury thermometer,
243

Steam engine, 106

Stefan's Law, 238

Sublimation, 163

Sun, temperature of, 250
Super-cooled liquid, 137
Surface tension, 66, 184, 264

Temperature, 2, 24 ; absolute gas,
80 ; scales of, 13, 22, 253, 316 ;
on the thermodynamic abso-
lute scale, definition of, 320

Thermal capacity, 87

Thermal efficiency, 119

Thermal conductivity, 189 ; of
common substances, 193

Thermal units, 86

Thermometer, 10 et seq.

Thermo-electric pyrometer, 245

Thermopile, 4

Thermos flask, 230

Thermo-syphon principle, 203, 204

Thomson, James, 275 ; William,
see Kelvin

Tobin tubes, 207

Trade Winds, 208

Units of heat, 86, 128
Ultra violet radiation, 227

Vacuum flask, 230

VanderWaals, 271

Vapour pressure, 156 ; at high

temperatures, 162 ; at curved

surfaces, 183 ; thermometers

243

Ventilation, 207
Volume change on melting, 129 ;

on vaporization, 147

Wave length, 224

Weight thermometer, 57

Wet and dry bulb hygrometer,

177

Wiedemann Franz Law, 194
Wien's Displacement Law, 240
Work done by engine, 108 ; by

gas expanding, 278, 285
Wroblewski, 289

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